Estimation of the conformal factor under bounded Willmore energy

Abstract

We prove for closed, orientable surfaces in $\ \mathbb{R }^3\$ with Willmore energy less that $\ 8 \pi - \delta \$ and whose conformal structures are compactly contained in moduli space that after applying appropriate Möbius transformations the conformal factors between the induced metrics and conformal metrics of constant curvature are uniformly bounded by constants depending only on $\ \delta > 0,$ the genus of the surfaces and the compact subset of the moduli space. Secondly, for a given sequence of closed, orientable surfaces as above, we prove that the conformal factor remains bounded without applying Möbius transformations if and only if no topology is lost. Similar estimates hold in higher codimension.

1 Introduction

For an immersion $\ f : {\Sigma }\rightarrow \mathbb{R }^n$ of a closed surface $\ {\Sigma }$, which we assume to be orientable, the Willmore functional is defined by

$$\begin{aligned} { \mathcal{W } }(f) = \frac{1}{4} \int \limits _{\Sigma }|\overrightarrow{\mathbf{H }}|^2 { \ \mathrm{{d}} }\mu _g, \end{aligned}$$

where $\ \overrightarrow{\mathbf{H }}$ denotes the mean curvature vector of $\ f, g = f^* g_{euc}$ the pull-back metric and $\ \mu _g$ the induced area measure on $\ {\Sigma }$. The main interest for the Willmore functional steams from its invariance under conformal transformations.

We continue the work of [9] and denote by $\ \beta _p^n$ the infimum of the Willmore energy of immersions $\ f: {\Sigma }\rightarrow \mathbb{R }^n$ of a closed, orientable surface $\ {\Sigma } \text{ of } \text{ genus } \,p$. We know $\ { \mathcal{W } }(f) \ge 4 \pi$ with equality only for round spheres, in particular $\ \beta _p^n \ge 4 \pi$. We put as in [9]

$$\begin{aligned} \tilde{\beta }^n_p := \min \left\{ 4 \pi +\sum _{i=1}^k (\beta ^n_{p_i}-4\pi ): 1 \le p_i < p,\,\sum _{i=1}^k p_i =p\right\} , \end{aligned}$$

(1.1)

where $\tilde{\beta }^n_1 = \infty$,

$$\begin{aligned} { e_{n} } := \left\{ \begin{array}{ll} 4 \pi &{} \quad \text{ for } \quad n = 3, \\ 8 \pi / 3 &{} \quad \text{ for } \quad n = 4, \\ 2 \pi &{} \quad \text{ for } \quad n \ge 5, \\ \end{array} \right. \end{aligned}$$

(1.2)

and define the constants

$$\begin{aligned} { \mathcal{W } }_{n,p} := \min ( 8 \pi , \tilde{\beta }^n_p , \beta ^n_p + { e_{n} } ). \end{aligned}$$

(1.3)

For $\ n = 3$, the last term could be ommitted as $\ \beta ^3_p + { e_{3} } > 8 \pi$.

By Poincaré’s theorem any smooth metric $\ g$ on $\ {\Sigma }\not \cong S^2$ is uniquely conformal to a unit volume constant curvature metric

$$\begin{aligned} { g_{{ poin }} }= e^{-2u} g. \end{aligned}$$

(1.4)

The compactness theorem [9] Theorem 4.1 or Theorem 5.3 in §5 for $\ n \ge 5$ estimates the conformal factor in (1.4) for the pull-back metric of a smooth immersion $\ f: {\Sigma }\rightarrow \mathbb{R }^n$ of a closed, orientable surface $\ {\Sigma } \text{ of } \text{ genus } \, p$ with

$$\begin{aligned} { \mathcal{W } }(f) \le { \mathcal{W } }_{n,p} - \delta \end{aligned}$$

(1.5)

after applying an appropriate Möbius transformation by

$$\begin{aligned} \parallel u \parallel _{L^\infty ({\Sigma })} \le C(n,p,\delta ). \end{aligned}$$

(1.6)

In the definition (1.3), the bound $\ 8 \pi$ excludes by the Li-Yau inequality in [11] branch points, and the bound $\ { \mathcal{W } }_{n,p} \le \tilde{\beta }_{n,p}$ prevents topological splitting in the sense that $\ p$ handles of $\ {\Sigma }\$ do not group in $\ p_1 \text{ and } \, p_2$ handles with $\ p_1 + p_2 = p, 1 \le p_1, p_2 < p$. The last bound $\ { \mathcal{W } }_{n,p} \le \beta _p^n + { e_{n} }$ is an algebraic condition in order to apply the Hardy space theory in the work of Müller and Sverak [12]. As $\ \beta _p^n \ge 4 \pi \text{ and } \,{ e_{3} } = 4 \pi$, this does not appear in (1.3) for $\ n = 3$. The constant $\ { e_{n} } = 2 \pi \text{ for } \,n \ge 5$ is directly taken from the general situation in [12], see §4. In [9] Theorem 6.1, the constants were adapted from [12] to our situation to $\ { e_{4} } = 8 \pi / 3$. Whether these constants are optimal for $\ n \ge 4\$ is not clear.

The compactness theorem [9] Theorem 4.1 was used in [10] to obtain existence of conformally constrained Willmore minimizers, these are minimizers of the Willmore energy of immersions conformal to a smooth metric $\ g_0 \text{ on }\, {\Sigma }$, when the infimum satisfies

$$\begin{aligned} { \mathcal{W } }({\Sigma },g_0,n) := \inf \{ { \mathcal{W } }(f)\ | \ f: {\Sigma }\rightarrow \mathbb{R }^{n} \text{ smooth } \text{ immersion } \text{ conformal } \text{ to } \, g_{0}\ \} < { \mathcal{W } }_{n,p}.\nonumber \\ \end{aligned}$$

(1.7)

The existence of conformally constrained Willmore minimizers was recently extended in [7] and [13] to $\ { \mathcal{W } }(\Sigma ,g_0,n) < 8 \pi \$ in any codimension. Moreover in [10] smoothness of any conformally constrained Willmore minimizer was shown.

Actually even if we do not fix the metric $\ g_0\$ or likewise the conformal class induced by $\ f$, the estimation of the conformal factor in (1.4) gives control on the pull-back metric after reparametrization, as it was shown in [9, Lemma 5.1] that a bound on the conformal factor and on the Willmore energy

$$\begin{aligned} { \mathcal{W } }(f), \max \limits _{\Sigma }|u| \le \Lambda \end{aligned}$$

implies that the induced conformal structure lie in a compact subset of the moduli space depending on $n,\Lambda \$ and the genus of ${\Sigma }$.

The first aim of this article is to prove a partial converse of [9, Lemma 5.1].

Theorem 1.1

Let $f: {\Sigma }\rightarrow \mathbb{R }^n\$ be a smooth immersion of a closed, orientable surface $\ {\Sigma }\text{ of } \text{ genus } \, p \ge 1\$ with

$$\begin{aligned} { \mathcal{W } }(f) \le \left\{ \begin{array}{l@{\quad }l} 8 \pi - \delta &{} \text{ for } \;\;\;n = 3, \\ \beta _p^n + { e_{n} } - \delta &{} \text{ for } \;\;\; n \ge 4, \\ \end{array} \right. \end{aligned}$$

for some $\ \delta > 0\$ and assume that the conformal structure induced by the pull-back metric of f lies in a compact subset K of the moduli space.

Then after applying an Möbius transformation, the pull-back metric $\ g := f^* { g_{euc} }\$ is uniformly conformal to a unit volume constant curvature metric $\ { g_{{ poin }} }:= e^{-2 u} g$, more precisely

$$\begin{aligned} \parallel u \parallel _{L^\infty ({\Sigma })} \le C(n,p,K,\delta ). \end{aligned}$$

$\square$

This converse is only partial as the energy bound is restricted by $8 \pi$ and the algebraic energy condition in (1.3). Here the bound ${ \mathcal{W } }_{n,p} \le \tilde{\beta }_{n,p}\$ is replaced by the compactness in moduli space, which is weaker by [7] Theorem 5.3 and Theorem 5.5 or [14] Theorem I.1.

The algebraic energy condition restricts the energy loss by comparing to the infimum $\ \beta _p^n$. This can be localized.

Theorem 1.2

Let $f: {\Sigma }\rightarrow \mathbb{R }^n$ be a smooth immersion of a closed, orientable surface $\ {\Sigma }\not \cong S^2\$ with

$$\begin{aligned} { \mathcal{W } }(f) \le \min (8 \pi , { \mathcal{W } }({\Sigma },f^* { g_{euc} },n) + { e_{n} } )- \delta \end{aligned}$$

for some $\delta > 0$. Moreover we assume that the conformal structure induced by the pull-back metric of f lies in a compact subset K of the moduli space.

Then after applying an Möbius transformation, the pull-back metric $\ g := f^* { g_{euc} }\$ is uniformly conformal to a unit volume constant curvature metric $\ { g_{{ poin }} }:= e^{-2 u} g$, more precisely

$$\begin{aligned} \parallel u \parallel _{L^\infty ({\Sigma })} \le C(n,p,K,\delta ). \end{aligned}$$

$\square$

Clearly Theorem 1.1 implies Theorem 1.2 as ${ \mathcal{W } }({\Sigma },g_0,n) \ge 4 \pi$. Theorem 1.1 is suited when working for example on a fixed Riemann surface. It extends the framework of [10] to ${ \mathcal{W } }(\Sigma ,g_0,n) < 8 \pi$ in any codimension.

There is a second aim of this article. The compactness theorem [9] Theorem 4.1 and Theorems 1.1 and 1.2 in this article all estimate the conformal factor after applying appropriate Möbius transformations. The dividing out the invariance group of the Willmore functional is certainly necessary, and it is sufficient to obtain existence results of conformally constrained Willmore minimizers. In applications, there is the stronger task to estimate the conformal factor for a given sequence without applying Möbius transformations. Therefore there is a need for a precise criterion, which can be checked on the given sequence, whether this sequence needs preparation by Möbius transformations or not. We recall that the bound $\ { \mathcal{W } }_{n,p} \le \tilde{\beta }_p^n\$ in (1.3) prevents topological splitting by an energy bound, and we actually see that preserving the topology is necessary and sufficient for the estimation of the conformal factor. To be more precise, we establish for a sequence of closed, orientable embedded surfaces $\ {\Sigma }_m \subseteq \mathbb{R }^n \text{ of } \text{ fixed } \text{ genus } \, p \ge 1\$ with

$$\begin{aligned} \begin{array}{c} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < 8 \pi ,\\ {\Sigma }_m \subseteq B_1(0), \\ { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }, \end{array} \end{aligned}$$

(1.8)

that $\ spt\ \mu \$ is a closed, orientable, embedded topological surface of $\ genus(spt\ \mu ) \le p$. Then without loss of topology in the sense that $\ genus(spt\ \mu ) = p\$ and with the algebraic energy condition, we prove an estimate of the conformal factor. In a second step, we replace no loss in topology by compactness of the conformal structures in moduli space and the non-triviality of the topology in the sense that $\ genus(spt\ \mu ) \ge 1$. Actually these conditions are equivalent.

Proposition 1.3

Let $\ {\Sigma }_m \subseteq \mathbb{R }^n\$ be closed, orientable, embedded surfaces of fixed genus $\ p \ge 1\$ with

$$\begin{aligned} \begin{array}{c} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < 8 \pi , \\ {\Sigma }_m \subseteq B_1(0), \nonumber \\ { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }. \end{array} \end{aligned}$$

Then $\ spt\ \mu \$ is a closed, orientable, embedded topological surface of $\ genus(spt\ \mu ) \le p$. No topology is lost in the sense that

$$\begin{aligned} genus(spt\ \mu ) = p \end{aligned}$$

if and only if some topology is kept in the sense that

$$\begin{aligned} genus(spt\ \mu ) \ge 1 \end{aligned}$$

and the conformal structues

$$\begin{aligned} { [{{\Sigma }_m}] }\,\, \text{ lie } \text{ in } \text{ a } \text{ compact } \text{ subset } \text{ of } \text{ the } \text{ moduli } \text{ space }. \end{aligned}$$

In this case if moreover

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < { \mathcal{W } }(\mu ) + { e_{n} }, \end{aligned}$$

then the induced metrics $\ g_m := { g_{euc} }| {\Sigma }_m\$ are uniformly conformal to unit volume constant curvature metrics $\ { g_{{ poin },m} }:= e^{-2 u_m} g_m\$ for $\ m\$ large, more precisely

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } \parallel u_m \parallel _{L^\infty (\Sigma )} < \infty . \end{aligned}$$

$\square$

We proceed from (1.8) and prove that some topology can always be kept in the sense of $\ genus(spt\ \mu ) \ge 1\$ after applying appropriate Möbius transformations. This yields Theorem 1.1 and in turn Theorem 1.2. Finally, we summarize the equivalence of no topological loss and the compactness in moduli space in the following theorem.

Proposition 1.4

Let $\ {\Sigma }_m \subseteq \mathbb{R }^n\$ be closed, orientable, embedded surfaces of genus $\ p \ge 1\$ with

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty }{ \mathcal{W } }({\Sigma }_m) < 8 \pi . \end{aligned}$$

Then the conformal structures induced by $\ {\Sigma }_m\ \$ lie in a compact subset of the moduli space if and only if no topology is lost after applying appropriate Möbius transformations, more precisely that any subsequence has a subsequence such that after applying appropriate Möbius transformations

$$\begin{aligned} { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures } \end{aligned}$$

with $\ spt\ \mu \$ is a closed, orientable, embedded topological surface and

$$\begin{aligned} genus(spt\ \mu ) = p. \end{aligned}$$

In this case after passing to a subsequence the conformal structures converge

$$\begin{aligned} { [{{\Sigma }_m}] } \rightarrow { [{spt\ \mu }] }\,\,\text{ in } \text{ moduli } \text{ space }. \end{aligned}$$

$\square$

The inclusion that compactness in moduli space keeps together the topology after applying appropriate Möbius transformations was already observed in [7], where uniformly conformal weak limits in $\ W^{2,2}_{loc}({\Sigma }- \mathcal{{S}}) \text{ for } \text{ some } \text{ finite } \,\mathcal{{S}} \subseteq {\Sigma }\$ were obtained.

2 Convergence without loss of topology

We start proving that measure theoretic limits under bounded Willmore energy have a topology and a genus.

Proposition 2.1

Let $\ {\Sigma }_m \subseteq \mathbb{R }^n\$ be closed, orientable, embedded surfaces of fixed genus $\ p \ge 0\$ with

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty }{ \mathcal{W } }({\Sigma }_m) < 8 \pi , \end{aligned}$$

(2.1)

$$\begin{aligned} \begin{array}{c} {\Sigma }_m \subseteq B_R(0), \\ { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }, \end{array} \end{aligned}$$

(2.2)

for some $\ R < \infty$.

Then $\ spt\ \mu \$ is a closed, orientable, embedded topological surface of genus

$$\begin{aligned} genus(spt\ \mu ) \le p. \end{aligned}$$

(2.3)

Moreover for a closed, orientable surface of same genus, there exists a uniformly conformal $\ W^{2,2}-$immersion, that is $\ f^* { g_{euc} }= e^{2u} g_0\$ for some smooth metric $\ g_0 \text{ on } \, {\Sigma } \text{ and } \,u \in L^\infty ({\Sigma })$, such that

$$\begin{aligned} f: {\Sigma }\, { \stackrel{\approx }{\longrightarrow } }\, spt \mu = f({\Sigma }) \end{aligned}$$

(2.4)

is a bi-lipschitz homeomorphism and for $\ \mu _f := f(\mu _g)\$

$$\begin{aligned} \mu _f = \mu = { \mathcal{H }^2 }\lfloor spt\ \mu . \end{aligned}$$

(2.5)

Proof

By (2.1) and lower semicontinuity

$$\begin{aligned} { \mathcal{W } }(\mu ) \le \limsup \limits _{m \rightarrow \infty }{ \mathcal{W } }({\Sigma }_m) < 8 \pi \end{aligned}$$

(2.6)

and by the Li-Yau inequality in [11] or [8] (A.17)

$$\begin{aligned} \theta ^2(\mu ,x) \le { \mathcal{W } }(\mu ) / (4 \pi ) < 2 \quad \text{ for } \text{ all } \quad x \in \mathbb{R }^n. \end{aligned}$$

We replace (2.1) by the weaker assumption

$$\begin{aligned} \begin{array}{c} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < \infty , \\ \theta ^2(\mu ,x) < 2 \quad \text{ for } \text{ all }\;\;\; x \in \mathbb{R }^n, \end{array} \end{aligned}$$

(2.7)

and proceed from here.

We may assume

$$\begin{aligned} { \mathcal{W } }({\Sigma }_m) \le { \mathcal{W } }- \delta \end{aligned}$$

(2.8)

for some $\ { \mathcal{W } }< \infty , \delta > 0\$ and get by lower semicontinuity

$$\begin{aligned} { \mathcal{W } }(\mu ) \le \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < \infty . \end{aligned}$$

(2.9)

By monotonicity formula, we get as in [16, Theorem 3.1]

$$\begin{aligned} {\Sigma }_m \rightarrow spt\ \mu \quad \text{ in } \text{ Hausdorff-distance }. \end{aligned}$$

(2.10)

Next by the Gauß equations and the Gauß-Bonnet theorem

$$\begin{aligned} \int \limits _{{\Sigma }_m} |A_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }= 4 { \mathcal{W } }({\Sigma }_m) + 8 \pi (p - 1) \le 4 { \mathcal{W } }+ 8 \pi (p - 1) =: A. \end{aligned}$$

(2.11)

Putting $\ \alpha _m := |A_{{\Sigma }_m}|^2 { \mathcal{H }^2 }\lfloor {\Sigma }_m$, we may assume after passing to a subsequence that $\ \alpha _m \rightarrow \alpha \$ weakly as Radon measures. Clearly $\ \alpha (\mathbb{R }^n) \le A < \infty$, and there are at most finitely many bad point $\ z_1, \ldots , z_N \in \mathbb{R }^n\$ with

$$\begin{aligned} \alpha (\{x_k\}) \ge 8 \pi - \delta \quad \text{ for }\quad k = 1, \ldots , N, \end{aligned}$$

(2.12)

for any fixed $0 < \delta < \pi \$ and $\ N \le A / (7 \pi ) =: N(p,{ \mathcal{W } })$. As $\alpha (B_\varrho (x) - \{x\}) \rightarrow 0 \text{ for } \,\varrho \rightarrow 0 \text{ and } \text{ any } \,x \in \mathbb{R }^n$, we can cover $\ { spt\ }\ \mu \subseteq \bigcup _{k=1}^K B_{\varrho _k/4}(x_k) \text{ with } \,x_k \in spt\ \mu \$ and

$$\begin{aligned} \alpha (B_{2 \varrho _k}(x_k) - \{x_k\}) < \varepsilon ^2 \quad \text{ for }\quad k = 1, \ldots , K \end{aligned}$$

(2.13)

and $\ \varepsilon \le \varepsilon _0(n,p,N,\delta )\$ small enough chosen below. Further we may assume that $\ x_k = x_k \text{ for } \,k = 1, \ldots , N\$ and

$$\begin{aligned} \alpha (B_{2 \varrho _k}(x_k)) < 8 \pi - \delta \quad \text{ for }\quad k = N + 1, \ldots , K. \end{aligned}$$

(2.14)

By (2.12) and (2.13), we see that $\ x_l \not \in B_{2 \varrho _k}(x_k), k \ne l = 1, \ldots , N$, hence

$$\begin{aligned} B_{\varrho _k}(x_k) \cap B_{\varrho _l}(x_l) = \emptyset \quad \text{ for }\quad k = 1, \ldots , N. \end{aligned}$$

(2.15)

As $\ \mu \ne 0$, we may further assume that

$$\begin{aligned} spt\ \mu \not \subseteq B_{2 \varrho _k}(x_k) \quad \text{ for }\quad k = 1, \ldots , K. \end{aligned}$$

(2.16)

Next by the monotonicity formula, see [16, 1.2] or [8, A.3 and A.5], writing $\perp$ for the projection onto $\ (T_x \mu )^\perp$,

$$\begin{aligned} \int \limits _{B_\varrho (x)} frac{|(\xi -x)^\perp |^2}{|\xi -x|^4} { \ \mathrm{{d}} }\mu (\xi ) < \infty \quad \text{ for } \text{ all } \quad x \in \mathbb{R }^n, \varrho > 0, \end{aligned}$$

and by (2.7), we can additionally assume for some $\ \theta ^2(\mu ,x_k) < \gamma _k < 2\$ that

$$\begin{aligned}&mu(\overline{B_{7 \varrho _k / 8}(x_k)}) < \gamma _k \pi (7 \varrho _k / 8)^2, \\&\displaystyle \int \limits _{B_{2 \varrho _k}(x_k)} \displaystyle {\frac{|(\xi -x_k)^\perp |^2}{|\xi -x_k|^4}} { \ \mathrm{{d}} }\mu (\xi ) < \varepsilon ^2. \end{aligned}$$

We get from above (2.2) and (2.7) for $\ 0 < r_m \ll \varrho _k/4, r_m \rightarrow 0\$ and $\ m\$ large enough that

$$\begin{aligned}&\displaystyle \int \limits _{{\Sigma }_m \cap B_{\varrho _k}(x_k) - B_{r_m}(x_k)} |A_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }< \varepsilon ^2, \nonumber \\&{ \mathcal{H }^2 }({\Sigma }_m \cap B_{7 \varrho _k/8}(x_k)) < \gamma _k \pi (7 \varrho _k / 8)^2, \nonumber \\&\displaystyle \int \limits _{{\Sigma }_m \cap B_{\varrho _k}(x_k) - B_{r_m}(x_k)} \displaystyle {\frac{|(\xi -x_k)^\perp |^2}{|\xi -x_k|^4}} { \ \mathrm{{d}} }{ \mathcal{H }^2 }(\xi ) < \varepsilon ^2, \end{aligned}$$

(2.17)

for $\ k = 1, \ldots , K$, and

$$\begin{aligned} \int \limits _{{\Sigma }_m \cap B_{\varrho _k}(x_k)} |A_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }< 8 \pi - \delta /2 \quad \text{ for }\quad k = N + 1, \ldots , K. \end{aligned}$$

(2.18)

By Hausdorff-convergence in (2.10) and $\ x_k \in spt\ \mu$, we get $\ {\Sigma }_m \cap B_{\varrho _k/4}(x_k) \ne \emptyset \text{ for } \,m\$ large. Moreover $\ {\Sigma }_m\$ is not contained in any $\ B_{\varrho _k}(x_k)\$ for $\ m\$ large by (2.16), and, as $\ {\Sigma }_m\$ is connected, any component of $\ {\Sigma }_m \cap B_{\sigma }(x_k), \sigma < 9 \varrho _k / 16$, extends to $\ \partial B_{9\varrho _k/16}(x_k)\$ in the sense of [9, Lemma 2.1 (a)]. Then we obtain as in [9, 4.16] for the multiplicity $\ M_k = m_k = 1\$ as in [9, 2.5], that is there is exactly one component of $\ D^k_m := D^k_{m,\sigma } \text{ of } \,{\Sigma }_m \cap B_{\sigma }(x_k) \text{ for } \,\sigma \in [5 \varrho _k/8, 7 \varrho _k/8]\$ appropriate as in Theorem 5.2 and that the multiplicity of its boundary entering in [9, 2.6] equals one in the sense

$$\begin{aligned} \left| \,\, \int \limits _{\partial D^k_m} k_{g_m} { \ \mathrm{{d}} }s - 2 \pi \right| \le C(n) \varepsilon ^{\alpha (n)} \end{aligned}$$

for appropriate $\ \alpha (n) > 0$. Denoting the genus of $\ D^k_m \oplus B_1(0) \text{ by } p_{m,k}$, we get by the Gauß-Bonnet theorem

$$\begin{aligned} \left| \,\, \int \limits _{D^k_m} K_{g_m} { \ \mathrm{{d}} }\mu _{g_m} + 4 \pi p_{m,k} \right| \le C(n) \varepsilon ^{\alpha (n)}. \end{aligned}$$

(2.19)

Passing to a subsequence, we may assume $\ p_k := p_{m,k}\$ is independent of $\ m\$ and after renumbering $\ x_k$, we may assume that

$$\begin{aligned} \left. \begin{array}{ll} &{} D^k_m \text{ is } \text{ not } \text{ a } \text{ disc, } \,p_k \ge 1, \\ \text{ or } \quad \\ &{} \int \nolimits _{{\Sigma }_m \cap B_{\varrho _k}(x_k)} |A_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }> 4 { e_{n} } - \delta , \\ \end{array} \right\} \quad \text{ for } \quad k = 1, \ldots , \tilde{N}, \end{aligned}$$

(2.20)

and

$$\begin{aligned} \left. \begin{array}{ll} D^k_m \text{ is } \text{ a } \text{ disc, } \, p_k = 0, \\ \int \nolimits _{{\Sigma }_m \cap B_{\varrho _k}(x_k)} |A_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }\le 4 { e_{n} } - \delta , \\ \end{array} \right\} \quad \text{ for } \quad k = \tilde{N} + 1, \ldots , K, \end{aligned}$$

(2.21)

when observing that (2.21) is certainly true for $\ k = N+1, \ldots , K\$ by (2.18), as $\ { e_{n} } \ge 2 \pi$, and

$$\begin{aligned} \int \limits _{\partial D^k_m} |K_{g_m}| { \ \mathrm{{d}} }\mu _{g_m} \le \frac{1}{2} \int \limits _{\partial D^k_m} |A_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }\mu _{g_m} \le 4 \pi - \delta / 4, \end{aligned}$$

hence $\ p_k = p_{m,k} = 0\$ by (2.19) for $\ C(n) \varepsilon ^{\alpha (n)} < \delta / 4$.

Then as in [9, Lemma 2.1 (b)], we may replace $\ {\Sigma }_m \text{ in } \,B_{M r_m}(x_k) \text{ for } \,k \!=\! 1, \ldots , \tilde{N} \text{ and } \,M\$ large observing (2.15) to obtain a closed, orientable, embedded surface $\ \tilde{\Sigma }_m\$ for $\ m \text{ large } \text{ and } \, M r_m \ll \varrho _k / 4\$ and

$$\begin{aligned}&{\Sigma }_m = \tilde{\Sigma }_m \quad \text{ in } \quad \mathbb{R }^n - \displaystyle \bigcup _{k=1}^{\tilde{N}} B_{M r_m}(x_k), \nonumber \\&\displaystyle \int \limits _{\tilde{\Sigma }_m \cap B_{\varrho _k}(x_k)} |A_{\tilde{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }< C(n) \varepsilon ^2 \quad \text{ for } \quad k = 1, \ldots , \tilde{N}, \\&\tilde{D}^k_m := \tilde{\Sigma }_m \cap B_\sigma (x_k) \text{ is } \text{ a } \text{ disc } \text{ for } \quad k = 1, \ldots , \tilde{N}.\nonumber \end{aligned}$$

(2.22)

Clearly

$$\begin{aligned} genus(\tilde{\Sigma }_m) = p - \sum \limits _{k=1}^{\tilde{N}} p_k \le p, \end{aligned}$$

(2.23)

and we choose ${\Sigma }\cong \tilde{\Sigma }_m$ with $0 \le genus({\Sigma }) \le p$.

Next

$$\begin{aligned} { \mathcal{W } }(\tilde{\Sigma }_m) \le { \mathcal{W } }({\Sigma }_m) + \sum \limits _{k=1}^{\tilde{N}} \frac{1}{4} \int \limits _{\tilde{\Sigma }_m \cap B_{M r_m}(x_k)} |\overrightarrow{\mathbf{H }}_{\tilde{\Sigma }_m}| { \ \mathrm{{d}} }{ \mathcal{H }^2 }\le { \mathcal{W } }({\Sigma }_m) + C(n) N \varepsilon ^2 \le { \mathcal{W } }- \delta / 2 \nonumber \\ \end{aligned}$$

(2.24)

by (2.8) for $\ \varepsilon = \varepsilon (n,N,\delta )\$ small enough. Again by the Li-Yau inequality in [11] or [8] (A.16)

$$\begin{aligned} |{ \mathcal{H }^2 }({\Sigma }_m \cap B_{M r_m}(x_k)) - { \mathcal{H }^2 }(\tilde{\Sigma }_m \cap B_{M r_m}(x_k))| \le C { \mathcal{W } }M^2 r_m^2 \quad \text{ for } \quad k = 1, \ldots , \tilde{N}, \end{aligned}$$

hence by (2.2)

$$\begin{aligned} { \mathcal{H }^2 }\lfloor \tilde{\Sigma }_m \rightarrow \mu \end{aligned}$$

(2.25)

and as in [16] Theorem 3.1

$$\begin{aligned} \tilde{\Sigma }_m \rightarrow spt\ \mu \quad \text{ in } \text{ Hausdorff-distance }. \end{aligned}$$

(2.26)

Further we see from (2.11) and (2.22) that

$$\begin{aligned} \int \limits _{\tilde{\Sigma }_m} |A_{\tilde{\Sigma }_m}| { \ \mathrm{{d}} }{ \mathcal{H }^2 }&\le \int \limits _{{\Sigma }_m} |A_{{\Sigma }_m}| { \ \mathrm{{d}} }{ \mathcal{H }^2 }+ \sum \limits _{k=1}^{\tilde{N}} \int \limits _{\tilde{\Sigma }_m \cap B_{M r_m}(x_k)} |A_{\tilde{\Sigma }_m}| { \ \mathrm{{d}} }{ \mathcal{H }^2 }\nonumber \\&\le A + C(n) N \varepsilon ^2 \le A + 1 \end{aligned}$$

for $\ \varepsilon = \varepsilon (n,N)\$ small enough, and putting $\ \tilde{\alpha }_m := |A_{\tilde{\Sigma }_m}|^2 { \mathcal{H }^2 }\lfloor \tilde{\Sigma }_m$, we get after passing to a subsequence that $\ \tilde{\alpha }_m \rightarrow \tilde{\alpha }\$ weakly as Radon measures and by (2.13), (2.21) and (2.22) that

$$\begin{aligned} \tilde{\alpha }(B_{2 \varrho _k}(x_k))&\le \alpha (B_{2 \varrho _k}(x_k) \!- B_{\varrho _k}(x_k)) \!+ C(n) \varepsilon ^2 \!\le C(n) \varepsilon ^2 \!< \pi \quad \text{ for } \quad k \!= 1, \!\ldots , \tilde{N}, \nonumber \\ \tilde{\alpha }(B_{2 \varrho _k}(x_k))&\le \alpha (B_{2 \varrho _k}(x_k) \!-\! B_{\varrho _k}(x_k)) \!+ 4 { e_{n} } \!- \delta < 4 { e_{n} } \!- \delta /2 \quad \text{ for } \, k \!=\! \tilde{N}\!+\!1, \!\ldots , K, \nonumber \\ \tilde{\alpha }(B_{2 \varrho _k}(x_k) \!- \{x_k\})&= \alpha (B_{2 \varrho _k}(x_k) \!- \{x_k\}) \!< \varepsilon ^2 \quad \text{ for } \, k \!= \tilde{N}\!+1, \!\ldots , K, \end{aligned}$$

(2.27)

for $\ \varepsilon = \varepsilon (n,N,\delta )\$ small enough. We get from (2.26) that $\ \tilde{\Sigma }_m \subseteq \bigcup _{k=1}^K B_{\varrho _k/2}(x_k) \text{ for } \,m\$ large and from (2.27)

$$\begin{aligned} \begin{array}{c} \displaystyle \int \limits _{\tilde{\Sigma }_m \cap B_{\varrho _k}(x_k)} |A_{\tilde{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }< 4 { e_{n} } - \delta /2, \nonumber \\ \displaystyle \int \limits _{\tilde{\Sigma }_m \cap B_{\varrho _k}(x_k) - B_{\varrho _k/2}(x_k)} |A_{\tilde{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }< C(n) \varepsilon ^2 \end{array} \end{aligned}$$

for $\ k = 1, \ldots , K$. For $\ \varepsilon \le \varepsilon (n,{ \mathcal{W } },\delta /2) / C(n)\$ as in Theorem 5.2, this verifies (5.9) and (5.10) for $\ \Lambda = { \mathcal{W } }, \delta /2 \text{ and } \,m\$ large. (5.11) is verified by (2.21) and (2.22), when observing (2.19). For $\ \tilde{\Sigma }_m \cong {\Sigma }\not \cong S^2$, we choose diffeomorphisms $\ \tilde{f}_m: {\Sigma }{ \stackrel{\approx }{\longrightarrow } }\tilde{\Sigma }_m\$ and conclude by [9] Theorem 3.1 or Theorem 5.2 for the pull-back metrics $\ \tilde{g}_m := \tilde{f}_m^* { g_{euc} }\$ and the conformal smooth unit volume constant curvature metrics $\ { \tilde{g}_{{ poin },m} }= e^{-2 \tilde{u}_m} \tilde{g}_m\$ that

$$\begin{aligned} \parallel \tilde{u}_m \parallel _{L^\infty ({\Sigma })} \le C(n, { \mathcal{W } }, \varrho _l / \varrho _k , K , p , \delta /2) \end{aligned}$$

(2.28)

for $\ m\$ large. Then by Proposition 6.1 after appropriate reparametrization of $\ \tilde{f}_m$, we get for a subsequence that

$$\begin{aligned} \begin{aligned}&\tilde{f}_m \rightarrow f \text{ weakly } \text{ in } \,W^{2,2}({\Sigma }), \text{ weakly }^* \text{ in } W^{1,\infty }({\Sigma }), \\&\tilde{u}_m \rightarrow u \text{ weakly } \text{ in } \, W^{1,2}({\Sigma }), \text{ weakly }^* \text{ in } L^\infty ({\Sigma }), \\&{ \tilde{g}_{{ poin },m} }\rightarrow { g_{{ poin }} }\quad \text{ smoothly }, \\&f^* { g_{euc} }= e^{2 u} { g_{{ poin }} }, \end{aligned} \end{aligned}$$

(2.29)

in particular $\ f\$ is a $\ W^{2,2}-$immersion uniformly conformal to $\ { g_{{ poin }} }$. By uniform convergence $\ \tilde{f}_m \rightarrow f$, bounded covergence $\ \sqrt{\tilde{g}_m} \rightarrow \sqrt{g}\$ and (2.25), we see

$$\begin{aligned} \mu _f := f(\mu _g) \leftarrow \tilde{f}_m(\mu _{\tilde{g}_m}) = { \mathcal{H }^2 }\lfloor \tilde{\Sigma }_m \rightarrow \mu . \end{aligned}$$

(2.30)

$\square$

We continue with our original assumption (2.1) instead of (2.7) and get by Proposition 7.2 and (2.6) that

$$\begin{aligned} { \mathcal{W } }(f) = { \mathcal{W } }(\mu ) < 8 \pi \end{aligned}$$

and

$$\begin{aligned} f:{\Sigma }\, { \stackrel{\approx }{\longrightarrow } }\, spt\ \mu = f({\Sigma }) \end{aligned}$$

is a bi-lipschitz homeomorphism and by (2.30)

$$\begin{aligned} \mu = \mu _f = { \mathcal{H }^2 }\lfloor spt\ \mu , \end{aligned}$$

which is (2.4) and (2.5), hence $\ spt\ \mu \cong {\Sigma }\$ is a closed, orientable, embedded topological surface of genus $\ \le p\$ by (2.23), which is as in (2.3), and the proposition is proved in the case that $\ {\Sigma }\not \cong S^2$.

In case $\ \tilde{\Sigma }_m \cong {\Sigma }\cong S^2$, we may assume after translation that $\ 0 \in \tilde{\Sigma }_m$. We select $\ \varrho > 0 \text{ with } \,\tilde{\alpha }(B_{2 \varrho }(0) - \{0\}) < \varepsilon ^2\$ and choose some $\ x_k = 0, \varrho _k = \varrho$. Then as in [9, Lemma 2.1 (b)], we replace $\ \tilde{\Sigma }_m \text{ in } \,B_{\varrho }(0) \text{ for } m\$ large to obtain a closed, orientable, embedded surface $\ S_m\$ with

$$\begin{aligned} \begin{array}{c} S_m = \tilde{\Sigma }_m \quad \text{ in } \mathbb{R }^n - B_{\varrho }(0), \\ \displaystyle \int \limits _{S_m \cap B_{\varrho }(0)} |A_{S_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }< C(n) \varepsilon ^2, \\ S_m = L_m \quad \text{ in } B_{\varrho /2}(0), \end{array} \end{aligned}$$

(2.31)

for some $\ 2-\text{ plane } L_m \ni 0$, which we may assume to be fixed $\ L = L_m\$ after suitable rotations. As the rotations and the above translations are compact, we still have (2.25) with a rotated and translated $\ \mu$. By the estimate in (2.31), we see that $\ S_m \cap B_\sigma (0)\$ is a disc for appropriate $\ \sigma = \sigma _{m,k} \in [5 \varrho / 8 , 7 \varrho / 8]$, hence recalling $\ \tilde{\Sigma }_m \cong S^2\$

$$\begin{aligned} S_m \cong S^2. \end{aligned}$$

(2.32)

Assuming by (2.1) that

$$\begin{aligned} { \mathcal{W } }({\Sigma }_m) \le 8 \pi - \delta \end{aligned}$$

for some $\ \delta > 0$, we get as in (2.24)

$$\begin{aligned} { \mathcal{W } }(S_m) \le { \mathcal{W } }({\Sigma }_m) + C(n) (N + 1) \varepsilon ^2 \le 8 \pi - \delta /2 \end{aligned}$$

for $\ \varepsilon \le \varepsilon (n,N,\delta )$. By the Li-Yau inequality in [11] or [8, A.16]

$$\begin{aligned} \varrho ^{-2} { \mathcal{H }^2 }(S_m \cap B_\varrho ) \le C, \end{aligned}$$

hence after passing to a subsequence

$$\begin{aligned} { \mathcal{H }^2 }\lfloor S_m \rightarrow \beta \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }. \end{aligned}$$

(2.33)

Then by lower semicontinuity and the Li-Yau inequality in [11] or [8, A.17]

$$\begin{aligned} \theta ^2(\beta ) \le { \mathcal{W } }(\beta ) / (4 \pi ) < 2. \end{aligned}$$

(2.34)

Now we take any orientable, connected surface $\ H \subseteq \mathbb{R }^2 \text{ with } H - B_{\varrho /4}(0) = L - B_{\varrho /4}(0)$, and which is not a disc, say

$$\begin{aligned} q := genus(H \cup \{\infty \}) \ge 1. \end{aligned}$$

(2.35)

We replace $\ S_m \text{ in } B_{\varrho /4}(0) \text{ by } H \cap B_{\varrho /4}(0)\$ to obtain a obtain a closed, orientable, embedded surface $\ \Gamma _m \not \cong S^2\$ with

$$\begin{aligned} \Gamma _m&= \tilde{\Sigma }_m \quad \text{ in } \mathbb{R }^n - B_{\varrho }(0), \nonumber \\ \Gamma _m&= S_m \quad \text{ in } \mathbb{R }^n - B_{\varrho /4}(0), \\ \Gamma _m&= H \quad \text{ in } B_{\varrho /2}(0).\nonumber \end{aligned}$$

(2.36)

Then clearly

$$\begin{aligned} \chi (\Gamma _m) = \chi (S_m - \overline{B_{\varrho /2}(0)}) + \chi (H \cap B_{\varrho /2}(0)), \end{aligned}$$

and by (2.32) and (2.35)

$$\begin{aligned} genus(\Gamma _m) = genus(H \cup \{\infty \}) = q \ge 1. \end{aligned}$$

(2.37)

As

$$\begin{aligned} { \mathcal{W } }(\Gamma _m) \le { \mathcal{W } }(S_m) + { \mathcal{W } }(H) \le 8 \pi + { \mathcal{W } }(H) < \infty \end{aligned}$$

(2.38)

is bounded, we get as above after passing to a subseqeuence

$$\begin{aligned} { \mathcal{H }^2 }\lfloor \Gamma _m \rightarrow \nu \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }. \end{aligned}$$

(2.39)

Clearly by (2.25), (2.33) and (2.36)

$$\begin{aligned} \nu \lfloor (\mathbb{R }^n - \overline{B_\varrho (0)})&= \mu \lfloor (\mathbb{R }^n - \overline{B_\varrho (0)}), \nonumber \\ \nu \lfloor (\mathbb{R }^n - \overline{B_{\varrho /4}(0)})&= \beta \lfloor (\mathbb{R }^n - \overline{B_{\varrho /4}(0)}), \\ \nu \lfloor B_{\varrho /2}(0)&= { \mathcal{H }^2 }\lfloor (H \cap B_{\varrho /2}(0)).\nonumber \end{aligned}$$

(2.40)

We see $\ \theta ^2(\nu ) = \theta ^2({ \mathcal{H }^2 }\lfloor H) \le 1 \text{ in } B_{\varrho /2}(0)\$ and by (2.34) that $\ \theta ^2(\nu ) = \theta ^2(\beta ) < 2 \text{ in } \mathbb{R }^n - \overline{B_{\varrho /4}(0)}$, hence combining

$$\begin{aligned} \theta ^2(\nu ,x) < 2 \quad \text{ for } \text{ all } \quad x \in \mathbb{R }^n. \end{aligned}$$

(2.41)

Therefore $\ \Gamma _m\$ satisfy with (2.38) and (2.41) the weaker assumption in (2.7), and we can proceed with $\ \Gamma _m\$ as in the beginning of the proof. As $\ \Gamma _m = H \text{ in } B_{\varrho /2}(0)\$ by (2.36) is smooth, we see that there are no bad points for $\ \Gamma _m \text{ in } B_{\varrho /2}(0)$. Then by (2.22) for $\Gamma _m$

$$\begin{aligned} \tilde{\Gamma }_m = \Gamma _m = H \quad \text{ in } B_{\varrho /2 - M r_m}(0) \end{aligned}$$

(2.42)

with $\ r_m \rightarrow 0$, and we conclude for $\ M r_m < \varrho /4\$ that

$$\begin{aligned} \chi (\tilde{\Gamma }_m) = \chi (\tilde{\Gamma }_m - \overline{B_{\varrho /4}(0)}) + \chi (H \cap B_{\varrho /4}(0)) \le 1 + \chi (H \cap B_{\varrho /4}(0)) = \chi (H \cap \{\infty \}), \end{aligned}$$

hence $\ genus(\tilde{\Gamma }_m) \ge genus(H \cup \{\infty \}) = q\$. As $\ genus(\tilde{\Gamma }_m) \le genus(\Gamma _m) = q\$ by (2.23) and (2.37), we get

$$\begin{aligned} genus(\tilde{\Gamma }_m) = q \ge 1, \end{aligned}$$

(2.43)

in particular $\ \Gamma \cong \tilde{\Gamma }_m\$ is not a sphere. Then by above there is a uniformly conformal $\ W^{2,2}-$immersion

$$\begin{aligned} h: \Gamma \,{ \stackrel{\approx }{\longrightarrow } }\, spt\ \nu \end{aligned}$$

(2.44)

which is a bi-lipschitz homeomorphism. Moreover by (2.37) and (2.43)

$$\begin{aligned} genus(spt\ \nu ) = genus(\Gamma ) = genus(\tilde{\Gamma }_m) = q = genus(H \cup \{\infty \}), \end{aligned}$$

and, as $\ spt\ \nu = H \text{ in } B_{\varrho /2}(0)\$ by (2.42), we conclude as above

$$\begin{aligned} \chi (spt\ \nu \!-\! \overline{B_{\varrho /3}(0)}) \!=\! \chi (spt\ \nu )\! - \chi (spt\ \nu \cap B_{\varrho /3}(0)) \!=\! \chi (spt\ \nu ) \!- (\chi (H \cup \{\infty \}) \!- 1) \!= 1, \end{aligned}$$

hence

$$\begin{aligned} spt\ \nu - \overline{B_{\varrho /3}(0)} \text{ is } \text{ a } \text{ topological } \text{ disc. } \end{aligned}$$

(2.45)

Next as $\ \mu = \nu \text{ in } \mathbb{R }^n - \overline{B_\varrho (0)}\$ by (2.40), we get that $\ { spt\ }\mu - \overline{B_\varrho (0)}\$ is an open, orientable, topological $\ 2-$manifold. As $\ \varrho \$ can be arbitrarily small and $\ 0 \in \tilde{\Sigma }_m\$ was arbitrary, $\ { spt\ }\mu \$ is an open, orientable, topological 2-manifold. Observing that $\ spt\ \mu \$ is compact and connected by (2.2), (2.10) and conectedness of $\ {\Sigma }_m$, we get

$$\begin{aligned} spt\ \mu \text{ is } \text{ a } \text{ closed, } \text{ orientable, } \text{ topological } \text{ surface }. \end{aligned}$$

We select an open neighbourhood $\ U(0) \text{ in } spt\ \mu \text{ of } 0\$ which is a disc and $\ \gamma := \partial U(p)\$ is a closed Jordan arc. For $\ \varrho \$ small enough, we have $\ \gamma \cap B_{2 \varrho }(0) = \emptyset \$ and

$$\begin{aligned} \gamma \subseteq spt\ \mu - \overline{B_\varrho (0)} = spt\ \nu - \overline{B_\varrho (0)} \subseteq spt\ \nu - \overline{B_{\varrho /3}(0)}. \end{aligned}$$

The last set is a disc by (2.45), hence the interior $\ I_\gamma \text{ of } \gamma \text{ in } spt\ \nu - \overline{B_{\varrho /3}(0)}\$ is a disc as well. Now $\ I_\gamma \$ is connected and

$$\begin{aligned} \partial I_\gamma \cap \partial B_\varrho (0) = \gamma \cap \partial B_\varrho (0) = \emptyset \end{aligned}$$

hence

$$\begin{aligned} I_\gamma \subseteq spt\ \nu - \overline{B_\varrho (0)} \quad \text{ or } \quad I_\gamma \subseteq spt\ \nu \cap B_\varrho (0). \end{aligned}$$

Observing that

$$\begin{aligned} \overline{I_\gamma } \cap spt\ \nu - \overline{B_\varrho (0)} \supseteq \gamma \ne \emptyset , \end{aligned}$$

we get

$$\begin{aligned} I_\gamma \subseteq spt\ \nu - \overline{B_\varrho (0)} = spt\ \mu - \overline{B_\varrho (0)}. \end{aligned}$$

We see that $\ I_\gamma \text{ and } U(0)\$ are open, closed and connected in $\ spt\ \mu - \gamma$, hence these are connetced components of $\ spt\ \mu - \gamma$. As $\ 0 \in U(0), 0 \not \in I_\gamma \$ by above and $\ spt\ \mu - \gamma \$ can have at most two components, we get

$$\begin{aligned} spt\ \mu = U(0) + \gamma + I_\gamma . \end{aligned}$$

(2.46)

Sicne both $\ U(0) \text{ and } I_\gamma \$ are discs, we get

$$\begin{aligned} spt\ \mu \cong S^2, \end{aligned}$$

which is (2.3) in the case $\ {\Sigma }\cong S^2$.

Finally by (2.44) and compactness of $\ spt\ \mu$, we get a finite atlas $\ \{ \varphi _1^{-1}, \ldots , \varphi _L^{-1} \}\$ of uniformal conformal $\ W^{2,2}$-immersions

$$\begin{aligned} \varphi _l: B_1(0) \; { \stackrel{\approx }{\longrightarrow } }\; U_l \subseteq spt\ \mu \quad \text{ for } \quad l = 1, \ldots , L, \end{aligned}$$

which are further bi-lipschitz homeomorphisms. This means in particular that

$$\begin{aligned} g_l := \varphi _l^* { g_{euc} }= e^{2 u_l} g_{0,l} \end{aligned}$$

for smooth metrics $\ g_l \text{ on } B_1(0) \text{ and } u_l \in L^\infty _{loc}(B_1(0))$. Introducing local conformal coordiantes for the smooth metrics $\ g_{0,l}$, we may assume after rearranging the atlas that $\ \varphi _l\$ are conformal with

$$\begin{aligned} g_l := \varphi _l^* { g_{euc} }= e^{2 u_l} { g_{euc} }\end{aligned}$$

and $\ u_l \in L^\infty (B_1(0))$. Then all transitions maps

$$\begin{aligned} \varphi _{k,l} := \varphi _k^{-1} \circ \varphi _l := \varphi _l^{-1}(U_l \cap U_k)\,\, { \stackrel{\approx }{\longrightarrow } }\,\,\varphi _k^{-1}(U_l \cap U_k) \end{aligned}$$

are conformal, hence holomorphic after possibly reversing the orientation, in particular smooth. As $\ u_l \in L^\infty (B_1(0))\$ and $\ D \varphi _{k,l}\$ is continuous, we see that $\ D \varphi _{k,l}\$ have full rank everywhere. Then $\ \varphi _{k,l}\$, being bijective, is a holomorphic diffeomorphism. With this atlas $\ spt\ \mu \$ is a compact, simply sonnected Riemann surface. By [6, Lemma 2.3.3], there exists a smooth conformal metric $\ g_0 \text{ on } spt\ \mu$, that is

$$\begin{aligned} \varphi _l: (B_1(0),{ g_{euc} })\; { \stackrel{\approx }{\longrightarrow } }\; (U_l,g_0) \end{aligned}$$

is conformal, and more precisely $\ \varphi _l^* g_0 = e^{2 v_l} { g_{euc} } \text{ for } \text{ some } v_l \in C^\infty (B_1(0))$, hence

$$\begin{aligned} \varphi _l^* g_0 = e^{2 v_l} { g_{euc} }= e^{2 (v_l - u_l)} \varphi _l^* { g_{euc} }\end{aligned}$$

and $\ g_0 = e^{2 (v_l - u_l) \circ \varphi _l^{-1}} { g_{euc} }$. As $\ u_l, v_l \in L^\infty _{loc}(B_1(0))$, we get

$$\begin{aligned} g_0 = e^{2 u_0} { g_{euc} }\quad \text{ on } spt\ \mu \text{ for } \text{ some } u_0 \in L^\infty (spt\ \mu ). \end{aligned}$$

(2.47)

We remark, since only $\ u_l \in L^\infty (B_1(0))$, we cannot conclude that $\ { g_{euc} }\$ is smooth on $\ spt\ \mu \$ with respect to the holomorphic atlas above.

By the uniformisation theorem for simply connected Riemann surfaces, see [4, Theorem IV.1.1], $\ (spt\ u,g_0)\$ is conformally equivalent to the sphere $\ S^2\$ with standard metric $\ g_{S^2} := { g_{euc} }| S^2$, hence there is a conformal diffeomorphism

$$\begin{aligned} f: (S^2,g_{S^2}) \; { \stackrel{\approx }{\longrightarrow } }\; (spt\ \mu , g_0), \end{aligned}$$

say

$$\begin{aligned} f^* g_0 = e^{2 v} g_{S^2} \quad \text{ for } \text{ some } v \in C^\infty (S^2). \end{aligned}$$

(2.48)

Now for any conformal chart $\ \psi : V\,\, { \stackrel{\approx }{\longrightarrow } }\,\, B_1(0) \text{ of } S^2 \text{ with } V \subseteq f_l^{-1}(U_l) \text{ for } \text{ some } l$, we see that

$$\begin{aligned} \psi _l := \varphi _l^{-1} \circ (f | V) \circ \psi ^{-1}: B_1(0) \rightarrow B_1(0) \end{aligned}$$

is conformal, hence smooth, and we conclude that $\ f \in W^{2,2}(S^2,\mathbb{R }^n)\$ and is lipschitz. Calculating the pull-back metrics by (2.47) and (2.48), we get

$$\begin{aligned} f^* { g_{euc} }= f^*(e^{-2 u_0} g_0) = e^{2 (v - (u_0 \circ f))} g_{S^2} \end{aligned}$$

with $\ v - (u_0 \circ f) \in L^\infty (S^2)$, hence $\ f \text{ is } \text{ a } W^{2,2}$-immersion uniformly conformal to $\ g_{S^2}$. Then by Proposition 7.2 (7.7)

$$\begin{aligned} \mu _f = f(\mu _g) = \# f^{-1} \cdot { \mathcal{H }^2 }\lfloor f(S^2) = { \mathcal{H }^2 }\lfloor spt\ \mu , \end{aligned}$$

as $\ f\$ is bijective.

Since $\ H_{{\Sigma }_m} \text{ is } \text{ bounded } \text{ in } L^2({ \mathcal{H }^2 }\lfloor {\Sigma }_m)\$ by (2.1), we get from (2.2) that $\ \mu \$ has weak mean curvature in $\ L^2(\mu )\$ and by Allard’s integral compactness theorem, see [1, Theorem 6.4] or [15, Remark 42.8], that $\ \mu \$ is an integral varifold. Moreover by (2.1), (2.2) and lower semicontinuity

$$\begin{aligned} { \mathcal{W } }(\mu ) \le \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < 8 \pi , \end{aligned}$$

hence by [8] (A.10) and (A.17) that $\ \theta ^2(\mu ) \ge 1 \text{ on } spt\ \mu \text{ and } \theta ^2(\mu ) \le { \mathcal{W } }(\mu ) / 4 \pi < 2$, hence $\ \mu \$ has unit density $\ \mu -$almost everywhere and by above

$$\begin{aligned} \mu = { \mathcal{H }^2 }\lfloor spt\ \mu = \mu _f, \end{aligned}$$

hence (2.5). Then by Proposition 7.2 and above

$$\begin{aligned} { \mathcal{W } }(f) = { \mathcal{W } }(\mu ) < 8 \pi \end{aligned}$$

and

$$\begin{aligned} f: S^2 \,{ \stackrel{\approx }{\longrightarrow } }\, spt\ \mu = f(S^2) \end{aligned}$$

is a bi-lipschitz homeomorphism, which is (2.4) in the case that $\ {\Sigma }\cong S^2$, and the proposition is fully proved. $\square$

Remarks  1. The sphere case when $\ {\Sigma }\cong S^2\$ in the above proposition is more elaborate, since we cannot estimate the conformal factor in (2.28) by [9] Theorem 3.1 or Theorem 5.2 as in the case when $\ {\Sigma }\not \cong S^2\$ due to the presence of non-trivial conformal transformations on the sphere. For $\ n = 3$, this could be done for $\ { \mathcal{W } }({\Sigma }_m) \le 6 \pi \$ by [2] and [3].

• 2. For a second homeomorphism $\ \hat{f}: \hat{\Sigma }\,{ \stackrel{\approx }{\longrightarrow } }\, spt\ \mu \$ with $\ \hat{f}\$ a uniformly conformal $\ W^{2,2}-$immersion, say $\ \hat{f}^* { g_{euc} }= e^{2 \hat{u}} { \hat{g}_{{ poin }} }\$ for some smooth unit volume constant curvature metric $\ { \hat{g}_{{ poin }} } \text{ and } \,\hat{u} \in L^\infty (\hat{\Sigma })$, we see that $\ \phi := f^{-1} \circ \hat{f}: \hat{\Sigma }{ \stackrel{\approx }{\longrightarrow } }{\Sigma }\$ is a homeomorphism. Moreover as $\ f\$ is bi-lipschitz and $\ \hat{f}\$ is lipschitz, we get that $\ \phi \$ is lipschitz and calculate the pull-back metric with (2.29)

  $$\begin{aligned} e^{2 \hat{u}} { \hat{g}_{{ poin }} }= \hat{f}^* { g_{euc} }= \phi ^* f^* { g_{euc} }= \phi ^*(e^{2u} { g_{{ poin }} }) = e^{2 u \circ \phi } \phi ^* { g_{{ poin }} }. \end{aligned}$$

  We see that

  $$\begin{aligned} \phi : (\hat{\Sigma },{ \hat{g}_{{ poin }} }) \rightarrow ({\Sigma },{ g_{{ poin }} }) \end{aligned}$$

  is conformal, hence holomorphic or anti-holomorphic, in particular smooth. As $\ u \in L^\infty ({\Sigma }), \hat{u} \in L^\infty (\hat{\Sigma })\$ and $\ D \phi \$ is continuous, we see that $\ D \phi \$ hat full rank everywhere on $\ \hat{\Sigma }$, and, as $\ \phi \$ is bijective, we get that $\ \phi \$ is a diffeomorphism. Then the conformal structures induced by $\ f \text{ respectively } \, \hat{f}\$ coincide, and we can define the conformal structure induced by $\ spt\ \mu \$ by putting

  $$\begin{aligned} { [{spt\ \mu }] } := { [{({\Sigma }, f^* { g_{euc} })}] } = { [{(\hat{\Sigma }, \hat{f}^* { g_{euc} })}] }. \end{aligned}$$

  (2.49)

  $\square$

Certainly by Möbius invariance of the Willmore funcitonal, the limit can be a sphere, and most of the information would be lost. Therefore it is important to have a device to ensure, after appropriately applying Möbius transformations, the non-triviality of the limit which means for us to keep some topology. This was for example done in the existence proof of Willmore minimizers under fixed genus in [16, Lemma 4.1] to avoid round spheres. Other examples are the arrangement lemma in [9, Lemma 4.1] or the 3-points normalization lemma in [13, Lemma III.1]. Here we successively apply [9, Lemma 4.1] to keep some topology in the general situation of the previous proposition.

Proposition 2.2

Let $\ {\Sigma }_m \subseteq \mathbb{R }^n\$ be closed, orientable, embedded surfaces of genus $\ p \ge 1\$ with

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < 8 \pi . \end{aligned}$$

(2.50)

Then after applying appropriate Möbius transformations and passing to a subsequence

$$\begin{aligned} \begin{array}{c} {\Sigma }_m \subseteq B_1(0), \\ { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }, \end{array} \end{aligned}$$

(2.51)

with $\ spt\ \mu \$ is a closed, orientable, embedded topological surface and

$$\begin{aligned} genus(spt\ \mu ) \ge 1. \end{aligned}$$

(2.52)

Proof

We avoid the case $\ {\Sigma }\cong S^2\$ in the previous proposition by applying appropriate Möbius transformations as in [9, Lemma 4.1] and may assume that

$$\begin{aligned} \begin{array}{c} \displaystyle {\Sigma }_m \subseteq B_1(0), \\ \displaystyle \int \limits _{{\Sigma }_m \cap B_{\varrho _0}(x)} |A^0_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }\le \displaystyle {\frac{1}{2}} \int \limits _{{\Sigma }_m} |A^0_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }\quad \text{ for } \text{ all } \;\;\; x \in \mathbb{R }^n, \end{array} \end{aligned}$$

(2.53)

$\ \varrho _0 = \varrho _0(n,E) > 0\$ and where $\ E := \int _{{\Sigma }_m} |A^0_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }$. By (2.50) we may assume

$$\begin{aligned} { \mathcal{W } }({\Sigma }_m) < 8 \pi - \delta \end{aligned}$$

(2.54)

for some $\ \delta > 0\$ and get by the Li-Yau inequality in [11] or [8, A.16]

$$\begin{aligned} \varrho ^{-2} { \mathcal{H }^2 }({\Sigma }_m \cap B_\varrho ) \le C, \end{aligned}$$

hence after passing to a subsequence

$$\begin{aligned} { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }. \end{aligned}$$

Now by (2.53), we cannot have $\ {\Sigma }_m \subseteq B_{\varrho _0}(x) \text{ for } \text{ any } \,x \in \mathbb{R }^n$, as $\ E > 0$, since $\ {\Sigma }_m\$ are no round spheres. Therefore $\ diam\ {\Sigma }_m \ge \varrho _0\$ and $\ { \mathcal{H }^2 }({\Sigma }_m) \ge c_0 \varrho _0^2\$ by [16, Lemma 1.1]. As $\ {\Sigma }_m \subseteq B_1(0)\$ by (2.53), we have $\ \mu (\mathbb{R }^n) = \lim _{m \rightarrow \infty } { \mathcal{H }^2 }({\Sigma }_m) > 0$, in particular $\ \mu \ne 0\$, hence together with (2.53), we get (2.51). Then we have (2.1) and (2.2) for $\ R = 1$, and we proceed as in Proposition 2.1 and use the notation there.

By the Gauß equations and the Gauß-Bonnet theorem, we have with (2.54)

$$\begin{aligned} \int \limits _{{\Sigma }_m} |A^0_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }= 2 { \mathcal{W } }({\Sigma }_m) + 8 \pi (p-1) \in [8 \pi p, 8 \pi (p+1) - 2 \delta [. \end{aligned}$$

(2.55)

Extending $\ D^k_m = {\Sigma }_m \cap B_\sigma (x_k) \text{ outside } \, B_\sigma (x_k)\$ to a flat plane near infinity as in [9, Lemma 2.1 (b)], we obtain

$$\begin{aligned} \left| \,\,\int \limits _{D^k_m} |A_{{\Sigma }_m}^0|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }- 2 { \mathcal{W } }(D^k_m) - 8 \pi p_k \right| \le C(n) \varepsilon ^{\alpha (n)}. \end{aligned}$$

Observing by extension to the inside as in (2.22) that

$$\begin{aligned} { \mathcal{W } }({\Sigma }_m - D^k_m) \ge 4 \pi - C(n) \varepsilon ^{\alpha (n)}, \end{aligned}$$

hence by (2.54)

$$\begin{aligned} { \mathcal{W } }(D^k_m) = { \mathcal{W } }({\Sigma }_m) - { \mathcal{W } }({\Sigma }_m - D^k_m) \le 4 \pi - \delta + C(n) \varepsilon ^{\alpha (n)}, \end{aligned}$$

(2.56)

we get for $\ C(n) \varepsilon ^{\alpha (n)} < \delta \$

$$\begin{aligned} \int \limits _{D^k_m} |A^0_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }\in ]8 \pi p_k - \delta , 8 \pi (p_k+1) - \delta [. \end{aligned}$$

(2.57)

Choosing $\ \varrho _k \le \varrho _0$, we get from (2.53), (2.55) and (2.57)

$$\begin{aligned} 8 \pi p_k - \delta < \int \limits _{{\Sigma }_m \cap B_{\varrho _k}(x)} |A^0_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }\le 4 \pi (p + 1) - \delta , \end{aligned}$$

hence

$$\begin{aligned} p_k < (p+1)/2 \le p \quad \text{ for } \quad k = 1, \ldots , \tilde{N}, \end{aligned}$$

(2.58)

and we see that one disc cannot take all the topology. When

$$\begin{aligned} spt\ \mu \cong S^2, \end{aligned}$$

(2.59)

we have $\ {\Sigma }\cong \tilde{\Sigma }_m \cong S^2$, and see from (2.23) that $\ \tilde{N} \ge 1\$ and after renumbering in (2.20) that

$$\begin{aligned} p_1 \ge 1. \end{aligned}$$

(2.60)

For $\ p = 1$, this is impossible by (2.58), and the proposition is proved for $p = 1$.

To proceed for $\ p \ge 2$, we do the replacement of $\ {\Sigma }_m\$ as in (2.22) but only in $\ B_{M r_m}(x_k) \text{ for } \,k = 2, \ldots , \tilde{N}\$ and obtain intermediate closed, orientable, embedded surfaces $\ \Gamma _m$. We get as in (2.24) for $\ { \mathcal{W } }= 8 \pi \$ that

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }(\Gamma _m) < 8 \pi \end{aligned}$$

and as in (2.25) and (2.26)

$$\begin{aligned} \begin{array}{c} { \mathcal{H }^2 }\lfloor \Gamma _m \rightarrow \mu , \\ \Gamma _m \rightarrow spt\ \mu \quad \text{ in } \text{ Hausdorff-distance }. \end{array} \end{aligned}$$

(2.61)

and calculate by our assumption $\ \tilde{\Sigma }_m \cong S^2$, (2.58) and (2.60) that

$$\begin{aligned} genus(\Gamma _m) = genus(\tilde{\Sigma }_m) + p_1 = p_1 \in \{ 1, \ldots , p-1 \}. \end{aligned}$$

Then by induction and compressing by a factor of $\ 1/2$, there are Möbius transformations $\ \Phi _m\$ such that after passing to a subseqeuence

$$\begin{aligned} \begin{aligned}&\Phi _m \Gamma _m \subseteq B_{1/2}(0), \\&{ \mathcal{H }^2 }\lfloor \Phi _m \Gamma _m \rightarrow \nu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }, \end{aligned} \end{aligned}$$

(2.62)

with $\ spt\ \nu \$ is a closed, orientable, embedded topological surface and

$$\begin{aligned} genus(spt\ \nu ) \ge 1. \end{aligned}$$

(2.63)

We claim

$$\begin{aligned} \begin{array}{c} \Phi _m {\Sigma }_m \subseteq B_1(0), \\ { \mathcal{H }^2 }\lfloor \Phi _m {\Sigma }_m \rightarrow \nu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }, \end{array} \end{aligned}$$

(2.64)

for $\ m\$ large, which yields (2.51) and (2.52) by (2.63).

To prove (2.64), we work with Hausdorff convergence and get from (2.10) for $\ \Phi _m \Gamma _m\$ that

$$\begin{aligned} \Phi _m \Gamma _m \rightarrow spt\ \nu \quad \text{ in } \text{ Hausdorff-distance }. \end{aligned}$$

(2.65)

We claim

$$\begin{aligned} \Phi _m {\Sigma }_m \rightarrow spt\ \nu \quad \text{ in } \text{ Hausdorff-distance }. \end{aligned}$$

(2.66)

If this is true, we see $\ \Phi _m {\Sigma }_m \subseteq B_1(0) \text{ for } \,m\$ large, as $\ spt\ \nu \subseteq \overline{B_{1/2}(0)}\$ by (2.62) and (2.65), which is the first part in (2.64). As $\ { \mathcal{W } }(\Phi _m {\Sigma }_m) = { \mathcal{W } }({\Sigma }_m) < 8 \pi - \delta \$ by (2.54), we get by the Li-Yau inequality in [11] or [8] (A.16) after passing to a subsequence

$$\begin{aligned} { \mathcal{H }^2 }\lfloor \Phi _m {\Sigma }_m \rightarrow \beta \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }. \end{aligned}$$

As $\ diam(spt\ \nu ) > 0$, we get $\ diam(\Phi _m {\Sigma }_m) \ge \varrho \text{ for } \text{ some } \,\varrho > 0$, hence $\ { \mathcal{H }^2 }(\Phi _m {\Sigma }_m) \ge c_0 \varrho ^2\$ by [16] Lemma 1.1. As $\ \Phi _m {\Sigma }_m \subseteq B_1(0)\$ by above, we have $\ \beta (\mathbb{R }^n) = \lim _{m \rightarrow \infty } { \mathcal{H }^2 }(\Phi _m {\Sigma }_m) > 0$, in particular $\ \beta \ne 0$. Therefore we have (2.1) and (2.2) for $\ \Phi _m {\Sigma }_m\$ and proceed as in Proposition 2.1. Then $\ \Phi _m {\Sigma }_m \rightarrow spt\ \beta \$ in Hausdorff-distance by (2.10), hence $\ spt\ \beta = spt\ \nu \$ by (2.66). We get by (2.5)

$$\begin{aligned} \beta = { \mathcal{H }^2 }\lfloor spt\ \beta = { \mathcal{H }^2 }\lfloor spt\ \nu = \nu , \end{aligned}$$

and (2.64) follows.

It remains to establish (2.66). We first prove

$$\begin{aligned} \Phi _m(\infty ) \not \rightarrow \infty . \end{aligned}$$

(2.67)

Otherwise we rotate under sterographic projection $\ \mathbb{R }^n \cup \{ \infty \} \cong S^n\$ slightly from $\ \Phi _m(\infty ) \text{ to } \,\infty \$ and get Möbius transformations $\ \Psi _m\$ with

$$\begin{aligned} \begin{array}{c} \Psi _m( \Phi _m(\infty ) ) = \infty , \\ \Psi _m \rightarrow id_{\mathbb{R }^n} \quad \text{ smoothly } \text{ on } \text{ compact } \text{ subsets } \text{ of } \,\mathbb{R }^n. \end{array} \end{aligned}$$

Then by (2.62) and (2.65)

$$\begin{aligned} \begin{array}{c} \Psi _m \Phi _m \Gamma _m \subseteq B_{3/4}(0) \quad \text{ for } \ m \ \text{ large },\\ { \mathcal{H }^2 }\lfloor \Psi _m \Phi _m \Gamma _m \rightarrow \nu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }, \\ \Psi _m \Phi _m \Gamma _m \rightarrow spt\ \nu \quad \text{ in } \text{ Hausdorff-distance }. \end{array} \end{aligned}$$

As $\ \Psi _m \Phi _m(\infty ) = \infty$, we see that $\ \Psi _m \Phi _m\$ is an isometry multiplied by some factor $\ \lambda _m > 0\$. Then by (2.61) and above

$$\begin{aligned} \lambda _m = { \mathcal{H }^2 }(\Psi _m \Phi _m \Gamma _m) / { \mathcal{H }^2 }(\Gamma _m) \rightarrow \nu (\mathbb{R }^n) / \mu (\mathbb{R }^n) =: \lambda \in ]0,\infty [, \end{aligned}$$

and we see for the isometries $\ { U }_m := \lambda _m^{-1} \Psi _m \Phi _m\$ that

$$\begin{aligned} { U }_m \Gamma _m \rightarrow \lambda ^{-1} spt\ \nu \quad \text{ in } \text{ Hausdorff-distance }. \end{aligned}$$

We select any $\ x_m \in \Gamma _m\$ and get by (2.61) after passing to a subsequence that $\ x_m \rightarrow x \in spt\ \mu \text{ and } \,{ U }_m x_m \rightarrow y \in \lambda ^{-1} spt\ \nu$. We concldue that

$$\begin{aligned} |{ U }_m(0)| \le |{ U }_m(x_m)| + |{ U }_m(x_m) - { U }_m(0)| = |{ U }_m(x_m)| + |x_m| \rightarrow |y| + |x|, \end{aligned}$$

hence after passing to a subsequence $\ { U }_m \rightarrow { U }$. Then by above and (2.61)

$$\begin{aligned} { U }(spt\ \mu ) \leftarrow { U }_m \Gamma _m \rightarrow \lambda ^{-1} spt\ \nu , \end{aligned}$$

hence $\ spt\ \mu \cong spt\ \nu$. This contradicts (2.59) and (2.63), hence we get (2.67).

From (2.67), we get after passing to a subsequence that $\ \Phi _m(\infty ) \rightarrow b \in \mathbb{R }^n$, and (2.62) is true for $\ \Phi _m \text{ replaced } \text{ by } \,\Phi _m - \Phi _m(\infty ) \text{ and } \,\nu \$ replaced by its translation $\ (x \mapsto x - b)_\# \nu$, in particular we can assume $\ \Phi _m(\infty ) = 0$. Let $\ I\$ be the inversion at the unit sphere, that is $\ I(x) := x / |x|^2$, then $\ I \circ \Phi _m\$ are Möbius transformations with $\ I \Phi _m(\infty ) = \infty$, hence are isometries multiplied by a factor $\ \lambda _m > 0\$ and

$$\begin{aligned} I \Phi _m(x) = \lambda _m O_m(x - a_m) \quad \text{ for } \text{ all } \ x \in \mathbb{R }^n \end{aligned}$$

and some $\ a_m \in \mathbb{R }^n\$ and some orthogonal $\ O_m$. After passing to a subsequence we get $O_m \rightarrow O$, and (2.62) remains true for $\ \Phi _m \text{ replaced } \text{ by } \,O_m^T \Phi _m \text{ and } \,\nu \$ replaced by its rotation $\ O^T_\# \nu$. Since $\ O_m^T I = I O_m^T \text{ and } \,I^{-1} = I$, we calculate

$$\begin{aligned} O_m^T \Phi _m(x) = I(\lambda _m (x - a_m)) \quad \text{ for } \text{ all } \quad x \in \mathbb{R }^n, \end{aligned}$$

hence we may assume

$$\begin{aligned} \Phi _m(x) = I(\lambda _m (x - a_m)) \quad \text{ for } \text{ all } \ x \in \mathbb{R }^n. \end{aligned}$$

(2.68)

If $\ 0 \not \in spt\ \nu$, then $\ { spt\ }\ \nu \subseteq \overline{B_{1/2}(0)} - B_\varepsilon (0) \text{ for } \text{ some } \, \varepsilon > 0\$ by (2.62), and we get that (2.62) remains true for $\ \Phi _m \text{ replaced } \text{ by } \,I \Phi _m \text{ and } \,\nu \$ replaced by its inversion $\ I_\# \nu$. Then we have $\ I \Phi _m(\infty ) = \infty$, which was already excluded in (2.67).

Therefore $\ 0 \in spt\ \nu$, and we select by (2.65) and $\ spt\ \nu \ne \{0\}\$ by (2.63) sequences $\ x_m, y_m \in \Gamma _m \text{ with } \,x_m \rightarrow x \ne 0, y_m \rightarrow 0\$. Then clearly by (2.68)

$$\begin{aligned} I(x_m), I(y_m) \in I \Phi _m \Gamma _m = \lambda _m (\Gamma _m - a_m), \end{aligned}$$

hence

$$\begin{aligned} \infty \leftarrow |I(x_m) - I(y_m)| \le diam(\lambda _m (\Gamma _m - a_m)) = \lambda _m diam(\Gamma _m). \end{aligned}$$

By (2.61), we see $\ diam(\Gamma _m) \rightarrow diam(spt\ \mu ) < \infty \$ and conclude $\ \lambda _m \rightarrow \infty$. From (2.68), we calculate the modulus of the derivative

$$\begin{aligned} |D \Phi _m(x)| = \lambda _m |DI(\lambda _m (x-a_m))| = \frac{1}{\lambda _m |x - a_m|^2} \quad \text{ for }\quad x \in \mathbb{R }^n. \end{aligned}$$

(2.69)

For a subsequence, we may assume that $\ a_m \rightarrow a \in \mathbb{R }^n \cup \{\infty \}\$ and get

$$\begin{aligned} |D \Phi _m| \rightarrow 0 \quad \text{ uniformly } \text{ on } \text{ compact } \text{ subsets } \text{ of }\, \mathbb{R }^n - \{ a \}. \end{aligned}$$

Now we are coming back to our construction from (2.22) and see

$$\begin{aligned} {\Sigma }_m&= \tilde{\Sigma }_m \quad \text{ in } \,\mathbb{R }^n - \bigcup _{k=1}^{\tilde{N}} B_{M r_m}(x_k), \nonumber \\ {\Sigma }_m&= \Gamma _m \quad \text{ in } \,\mathbb{R }^n - \bigcup _{k=2}^{\tilde{N}} B_{M r_m}(x_k), \\ \Gamma _m&= \tilde{\Sigma }_m \quad \text{ in } \, \mathbb{R }^n - B_{M r_m}(x_1).\nonumber \end{aligned}$$

(2.70)

If $\ a \ne x_1$, then

$$\begin{aligned} diam(\Phi _m(B_{M r_m}(x_1))) \rightarrow 0 \end{aligned}$$

and $\ \Phi _m \Gamma _m \text{ and } \,\Phi _m \tilde{\Sigma }_m\$ have the same limit in Hausdorff-distance, hence by (2.65)

$$\begin{aligned} \Phi _m \tilde{\Sigma }_m \rightarrow spt\ \nu \quad \text{ in } \text{ Hausdorff-distance }. \end{aligned}$$

Then as above by the Li-Yau inequality and (2.5)

$$\begin{aligned} { \mathcal{H }^2 }\lfloor \Phi _m \tilde{\Sigma }_m \rightarrow \nu \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }. \end{aligned}$$

Next $\ \limsup _{m \rightarrow \infty } { \mathcal{W } }(\tilde{\Sigma }_m) < 8 \pi \$ by (2.24) for $\ { \mathcal{W } }= 8 \pi$, and we conclude by Proposition 2.1

$$\begin{aligned} genus(spt\ \nu ) \le genus(\tilde{\Sigma }_m) = genus(spt\ \mu ), \end{aligned}$$

which contradicts (2.59) and (2.63), hence we get $\ a = x_1$.

Then as above $\ \Phi _m \Gamma _m \text{ and } \,\Phi _m {\Sigma }_m\$ have the same limit in Hausdorff-distance, hence by (2.65)

$$\begin{aligned} \Phi _m {\Sigma }_m \rightarrow spt\ \nu \quad \text{ in } \text{ Hausdorff-distance }, \end{aligned}$$

which is (2.66), and the proposition is proved. $\square$

Remark We strengthen (2.56) to

$$\begin{aligned} { \mathcal{W } }(D^k_m)&\ge \beta _{p_k}^n - 4 \pi - C(n) \varepsilon ^{\alpha (n)}, \nonumber \\ { \mathcal{W } }({\Sigma }_m - \cup _{k=1}^{\tilde{N}} D^k_m)&\ge \beta _{genus({\Sigma })}^n - C(n) \varepsilon ^{\alpha (n)}, \end{aligned}$$

hence

$$\begin{aligned} { \mathcal{W } }({\Sigma }_m)&= { \mathcal{W } }\left( {\Sigma }_m - \cup _{k=1}^{\tilde{N}} D^k_m\right) + \sum \limits _{k=1}^{\tilde{N}} { \mathcal{W } }(D^k_m) \nonumber \\&\ge \beta _{genus({\Sigma })}^n - C(n) \varepsilon ^{\alpha (n)} + \sum \limits _{k=1}^{\tilde{N}} (\beta _{p_k}^n - 4 \pi - C(n) \varepsilon ^{\alpha (n)})\nonumber \\&\ge \beta _{genus({\Sigma })}^n + \sum \limits _{k=1}^{\tilde{N}} (\beta _{p_k}^n - 4 \pi ) - C(n,N) \varepsilon ^{\alpha (n)}). \end{aligned}$$

Now

$$\begin{aligned} p = genus({\Sigma }) + \sum \limits _{k=1}^{\tilde{N}} p_k \end{aligned}$$

by (2.23), and if the limit keeps some topology in the sense that $\ genus({\Sigma }) = genus(spt\ \mu ) \ge 1\$ then $\ p_k < p$, and we conclude

$$\begin{aligned} 1 \le genus({ spt\ }\ \mu ) < p \Longrightarrow { \mathcal{W } }({\Sigma }_m) \ge \tilde{\beta }_p^n - C(n,N) \varepsilon ^{\alpha (n)} \end{aligned}$$

for the constant defined in (1.1).

We conclude, if we strengthen (2.1) to

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < \min (8 \pi , \tilde{\beta }_p^n), \end{aligned}$$

(2.71)

then we have after applying appropriate Möbius transformations that no topology is lost in the sense that

$$\begin{aligned} genus(spt\ \mu ) = p. \end{aligned}$$

$\square$

Estimation of the conformal factor is only possible, if no topology is lost. In the next proposition, we prove that if no topology is lost then the induced conformal structures lie in a compact subset of moduli space. Actually we do not need this proposition for the estimation of the conformal factor, but it will give together with the following section the equivalence of no topological loss after applying appropriate Möbius transformations and compactness in moduli space.

Proposition 2.3

If in Proposition 2.1 no topology is lost in the sense that

$$\begin{aligned} genus(spt\ \mu ) = p \ge 1, \end{aligned}$$

(2.72)

then the conformal structures induced by $\ {\Sigma }_m\$ lie in a compact subset of the moduli space.

Proof

We proceed with the notation of Proposition 2.1. If no topology is lost as in (2.72), we get from (2.23) that

$$\begin{aligned} p_k = 0 \quad \text{ for } \quad k = 1, \ldots , K, \end{aligned}$$

(2.73)

in particular $\ D^k_m \cong \tilde{D}^k_m\$ are discs, and by construction in (2.22) there are diffeomorphisms

$$\begin{aligned} \begin{array}{c} \psi _m: \tilde{\Sigma }_m { \stackrel{\approx }{\longrightarrow } }{\Sigma }_m, \\ \psi _m = id \quad \text{ on } \,{\Sigma }_m - \cup _{k=1}^{\tilde{N}} B_{M r_m}(x_k), \\ \psi _m(\tilde{\Sigma }_m \cap B_{M r_m}(x_k)) \subseteq B_{M r_m}(x_k). \end{array} \end{aligned}$$

(2.74)

We define $\ f_m := \psi _m \circ \tilde{f}_m: {\Sigma }\; { \stackrel{\approx }{\longrightarrow } }\; {\Sigma }_m, g_m := f_m^*{ g_{euc} }\$ and consider the unit volume constant curvature metrics $\ { g_{{ poin },m} }= e^{-2u_m} g_m$. By (2.4) and $\ x_k \in spt\ \mu = f({\Sigma })$, we define $\ q_k := f^{-1}(x_k)\$ and see for any $\ \Omega ^{\prime } \subset \subset \Omega := {\Sigma }- \{ q_1, \ldots , q_{\tilde{N}} \}\$ by uniform convergence in (2.29) and $\ r_m \rightarrow 0\$ that $\ d(\tilde{f}_m(\Omega ^{\prime }),x_k) > M r_m$, hence $\ \tilde{f}_m(\Omega ^{\prime }) \cap B_{M r_m}(x_k) = \emptyset$, and

$$\begin{aligned} f_m = \tilde{f}_m, g_m = \tilde{g}_m \quad \text{ on } \,\Omega ^{\prime } \end{aligned}$$

(2.75)

for $\ m\$ large depending on $\ \Omega ^{\prime }$, in particular by (2.29)

$$\begin{aligned} g_m \rightarrow f^* { g_{euc} }= e^{2u} { g_{{ poin }} }\quad \text{ pointwise } \text{ in } \,\Omega . \end{aligned}$$

By elementary differential geometry, we know

$$\begin{aligned} - \Delta _{g_m} u_m + 2 \pi \chi ({\Sigma }) e^{-2u_m} = K_{g_m}, \quad - \Delta _{\tilde{g}_m} \tilde{u}_m + 2 \pi \chi ({\Sigma }) e^{-2 \tilde{u}_m} = K_{\tilde{g}_m} \end{aligned}$$

on $\ {\Sigma }$, hence

$$\begin{aligned} \!-\! \Delta _{ \tilde{g}_{{ poin },m} }(u_m - \tilde{u}_m) \!=\! - e^{2 \tilde{u}_m} \Delta _{\tilde{g}_m}(u_m \!-\! \tilde{u}_m) \!=\! - 2 \pi \chi ({\Sigma }) (e^{-2 (u_m \!-\! \tilde{u}_m)} \!-\! 1) \quad \text{ in } \,\Omega ^{\prime },\quad \end{aligned}$$

(2.76)

and, as $\ \chi ({\Sigma }) = 2 (1 - p) \le 0$,

$$\begin{aligned} \Delta _{ \tilde{g}_{{ poin },m} }(\tilde{u}_m - u_m)_+ \ge 0 \quad \text{ in } \,\Omega ^{\prime } \end{aligned}$$

for $\ m\$ large. For any $\ q \in \Omega \$ there exists an open neighbourhood $\ U(q) \subset \subset \Omega \text{ of } \,q\$ which is a disc by $\ \varphi : U(q) \cong B_1(0)$. Then by local maximum estimates, see [5, Theorem 8.17], (2.29) and writing $\ \varphi : U_\varrho (q) \cong B_\varrho (0)\$ that

$$\begin{aligned} \sup \limits _{U_{1/2}(q)} (\tilde{u}_m - u_m) \le C({ g_{{ poin }} },q)\parallel \!(\tilde{u}_m - u_m)_+\!\parallel _{L^2(U_{3/4}(q))} \end{aligned}$$

for $\ m\$ large. To estimate the norm on the right-side, we observe

$$\begin{aligned} 1&= \mu _{{ g_{{ poin },m} }}({\Sigma }) = \int \limits _{\Sigma }e^{-2 u_m} { \ \mathrm{{d}} }\mu _{g_m} \ge \int \limits _{U(q)} e^{-2 u_m} { \ \mathrm{{d}} }\mu _{\tilde{g}_m}\nonumber \\&= \int \limits _{U(q)} e^{2 (\tilde{u}_m - u_m)} { \ \mathrm{{d}} }\mu _{{ \tilde{g}_{{ poin },m} }} \ge \int \limits _{U(q)} (1 + \tilde{u}_m - u_m)_+^2 { \ \mathrm{{d}} }\mu _{ \tilde{g}_{{ poin },m} }\end{aligned}$$

and get using (2.29)

$$\begin{aligned} u_m - \tilde{u}_m \ge - C({ g_{{ poin }} },q) \quad \text{ on } \,U_{1/2}(q) \end{aligned}$$

(2.77)

for $\ m\$ large.

Then by (2.76)

$$\begin{aligned} | \Delta _{ \tilde{g}_{{ poin },m} }(u_m - \tilde{u}_m) | \le C({ g_{{ poin }} },p,q) \quad \text{ in } \,U_{1/2}(q), \end{aligned}$$

and $\ u_m - \tilde{u}_m - \Gamma \ge 0 \text{ on } \,U_{1/2}(q) \text{ for } \,\Gamma = C({ g_{{ poin }} },q)$, and we get by the Harnack inequality, see [5, Theorem 8.17 and 8.18], and (2.29)

$$\begin{aligned} \sup \limits _{U_{1/4}(q)} (u_m - \tilde{u}_m - \Gamma ) \le C({ g_{{ poin }} },q) \inf \limits _{U_{1/4}(q)} (u_m - \tilde{u}_m - \Gamma ) + C({ g_{{ poin }} },p,q) \end{aligned}$$

and

$$\begin{aligned} \sup \limits _{U_{1/4}(q)} (u_m - \tilde{u}_m) \le C({ g_{{ poin }} },q) \inf \limits _{U_{1/4}(q)} (u_m - \tilde{u}_m) + C({ g_{{ poin }} },p,q) \end{aligned}$$

(2.78)

for $\ m\$ large.

Now if $\limsup _{m \rightarrow \infty } \sup _{U_{1/4}(q)} u_m < \infty$, we see from (2.28) and (2.77) that $\ u_m\$ is bounded from above and below on $\ U_{1/4}(q)$. Otherwise for a subsequence $\ \sup _{U_{1/4}(q)} u_m \rightarrow \infty$. Then by (2.28) and (2.78)

$$\begin{aligned} \inf \limits _{U_{1/4}(q)} u_m \rightarrow \infty , \end{aligned}$$

hence $\ u_m \rightarrow \infty \text{ uniformly } \text{ on } \,U_{1/4}(q)$.

Covering appropriately, we get after passing to a subsequence either

$$\begin{aligned} \begin{array}{ll} &{} u_m \rightarrow \infty \text{ uniformly } \text{ on } \text{ compact } \text{ subsets } \text{ of } \,\Omega , \\ \text{ or } \\ &{} u_m \text{ is } \text{ uniformly } \text{ bounded } \text{ on } \text{ compact } \text{ subsets } \text{ of } \, \Omega . \\ \end{array} \end{aligned}$$

(2.79)

We define

$$\begin{aligned} { \ell }_m := \inf \{ l_{{ g_{{ poin },m} }}(\gamma )\ | \ \gamma \text{ is } \text{ a } \text{ closed } \text{ geodesic } \text{ in } \, ({\Sigma },{ g_{{ poin },m} })\} > 0, \end{aligned}$$

where $\ l_{{ g_{{ poin },m} }}\$ denotes the length with respect to $\ { g_{{ poin },m} }$. By the Mumford compactness theorem, see e.g. [17] Theorem C.1, the conformal structures induced by $\ { g_{{ poin },m} } \text{ respectively } \text{ by } \,g_m\$ lie in a compact subset of the moduli space, if

$$\begin{aligned} \liminf \limits _{m \rightarrow \infty } { \ell }_m > 0. \end{aligned}$$

(2.80)

Therefore we prove (2.80) and assume $\ { \ell }_m \le 1\$ in the following.

If (2.80) is not true, we get after passing to a subsequence that $\ { \ell }_m \rightarrow 0$. We first prove this implies the first case in (2.79), that is

$$\begin{aligned} u_m \rightarrow \infty \text{ uniformly } \text{ on } \text{ compact } \text{ subsets } \text{ of } \,\Omega . \end{aligned}$$

(2.81)

By definition of $\ { \ell }_m$, there exists a closed geodesic $\ \gamma _m \text{ in } \,({\Sigma },{ g_{{ poin },m} })$ of length ${ \ell }_m \le { \ell }_{m,0} = l_{{ g_{{ poin },m} }}(\gamma _m) \le 2 { \ell }_m$. Further $\ \gamma _m\$ is non-nullhomotopic, see [17, pp. 184–185].

We consider any non-nullhomotopic curve $\ \gamma _m \text{ in } \, {\Sigma }\$ with

$$\begin{aligned} { \ell }_{m,0} = l_{{ g_{{ poin },m} }}(\gamma _m) \rightarrow 0. \end{aligned}$$

(2.82)

As $\ D^k_m = B_\sigma (x_k) \cap {\Sigma }_m, \sigma \in [5 \varrho _k / 8 , 7 \varrho _k / 8]\$ appropriate, are discs by (2.73), we see that $\ \tilde{\gamma }_m := f_m(\gamma _m)\$ cannot stay in any $\ B_{5 \varrho _k/8}(x_k) \cap {\Sigma }_m$. As $\ {\Sigma }_m = f_m({\Sigma }) \subseteq \cup _{k=1}^K B_{\varrho _k/2}(x_k) \text{ for } \,m\$ large, we get after passing to a subsequence

$$\begin{aligned} \tilde{\gamma }_m \cap B_{\varrho _l / 2}(x_l) \ne \emptyset \quad \text{ for } \text{ some } \,l \in \{ 1, \ldots , K \} \end{aligned}$$

(2.83)

and

$$\begin{aligned} l_{g_m}(\gamma _m) \ge \varrho _l / 8 \quad \text{ for } \text{ some } \,l \in \{ 1, \ldots , K \} \end{aligned}$$

(2.84)

and $\ m\$ large. If

$$\begin{aligned} \tilde{\gamma }_m \cap B_{\varrho _k / 2}(x_k) = \emptyset \quad \text{ for } \text{ all } \,\quad k \in \{ 1, \ldots , \tilde{N} \} \end{aligned}$$

(2.85)

and $\ m\$ large, in particular $\ l \not \in \{ 1, \ldots , \tilde{N} \}$, we see by (2.74) and $\ M r_m \le \varrho _k\$ that $\ f_m = \tilde{f}_m \text{ on } \,\gamma _m\$ and by uniform convergence in (2.29)

$$\begin{aligned} \liminf \limits _{m \rightarrow \infty } d(f(\gamma _m) , x_k) \ge \varrho _k/2 \quad \text{ for } \text{ all } \,\quad k \in \{ 1, \ldots , \tilde{N} \}. \end{aligned}$$

Putting

$$\begin{aligned} \Omega _\tau := f^{-1} \left( \mathbb{R }^n - \cup _{k=1}^{\tilde{N}} \overline{B_{\tau \varrho _k}(x_k)} \right) \subset \subset \Omega \quad \text{ for }\quad 0 < \tau < 1, \end{aligned}$$

we see

$$\begin{aligned} \gamma _m \subseteq \Omega _{1/4} \subset \subset \Omega \quad \text{ for } \ m \; \text{ large }. \end{aligned}$$

(2.86)

We get from (2.82) and (2.84) that the first case in (2.79) is true, that is (2.81).

In case (2.85) is not true, we can choose $\ l \in \{ 1, \ldots , \tilde{N} \}\$ in (2.83). As $\ \tilde{\gamma }_m\$ cannot stay in $\ B_{5 \varrho _l / 8}(x_l)$, we get as in (2.84) a subarc $\ \gamma _{m,0} \text{ of } \,\gamma _m\$ whose image $\ \tilde{\gamma }_{m,0} := f_m(\gamma _{m,0})\$ has its endpoints in $\ \partial B_{\varrho _l/2}(x_l) \text{ and } \,\partial B_{5 \varrho _l / 8}(x_l)\$ and is contained in $\ \overline{B_{5 \varrho _l / 8}(x_l)} - B_{\varrho _l / 2}(x_l)$. Then we see by (2.15) that

$$\begin{aligned} \tilde{\gamma }_{m,0} \cap B_{\varrho _k}(x_k) = \emptyset \quad \text{ for } \text{ all } \quad k \ne l, k \in \{ 1, \ldots , \tilde{N} \}, \end{aligned}$$

hence as above

$$\begin{aligned} \gamma _{m,0} \subseteq \Omega _{1/4} \subset \subset \Omega \quad \text{ for } \;\;\; m \text{ large }. \end{aligned}$$

(2.87)

Clearly $\ l_{g_m}(\gamma _{m,0}) \ge \varrho _l / 8\$ and again by (2.82), we get that the first case in (2.79) is true, that is (2.81).

We conclude from (2.81) together with (2.28), (2.29) and (2.75) for any $\ \Omega ^{\prime } \subset \subset \Omega \$ that

$$\begin{aligned} diam(\Omega ^{\prime },{ g_{{ poin },m} })&\le e^{- \inf _{\Omega ^{\prime }} u_m} diam(\Omega ^{\prime },g_m) = e^{- \inf _{\Omega ^{\prime }} u_m} diam(\Omega ^{\prime },\tilde{g}_m) \nonumber \\&\le e^{- \inf _{\Omega ^{\prime }} u_m} e^{\sup _{{\Sigma }} \tilde{u}_m} diam({\Sigma },{ \tilde{g}_{{ poin },m} }) \le C e^{- \inf _{\Omega ^{\prime }} u_m} \rightarrow 0. \end{aligned}$$

(2.88)

For $\ p = 1$, that is $\ {\Sigma }_m\$ are tori, there exist lattices $\ \Gamma _m = \mathbb{Z }+ (a_m + ib_m) \mathbb{Z }\subseteq \mathbb{C } \text{ with } \, 0 \le a_m \le 1/2, b_m > 0, a_m^2 + b_m^2 \ge 1\$ and such that $\ ({\Sigma }_m,{ g_{{ poin },m} })\$ is isometric to $\ (\mathbb{C }/\Gamma _m,b_m^{-1} { g_{euc} })$, as $\ { g_{{ poin },m} }\$ have unit volume, see [6, §2.7]. Here we see $\ { \ell }_m = 1 / \sqrt{b_m}$, and at each point of $\ ({\Sigma }_m,{ g_{{ poin },m} })\$ there exists a non-nullhomotopic curve $\gamma _m$ of length ${ \ell }_m \rightarrow 0$. Then by (2.86) and (2.87), we get $\ \gamma _m \cap \Omega _{1/4} \ne \emptyset \text{ for } \,m\$ large, and we conclude by (2.88) that

$$\begin{aligned} diam({\Sigma }_m,{ g_{{ poin },m} }) \le { \ell }_m + diam(\Omega _{1/4},{ g_{{ poin },m} }) \rightarrow 0. \end{aligned}$$

But

$$\begin{aligned} diam({\Sigma }_m,{ g_{{ poin },m} }) = diam(\mathbb{C }/\Gamma _m,b_m^{-1} { g_{euc} }) \ge \sqrt{b_m} / 2 = 1 / (2 { \ell }_m) \rightarrow \infty , \end{aligned}$$

which is a contradiction, and the proposition is proved for $p = 1$.

For $\ p \ge 2$, we proceed similarly and obtain from the collar lemma, see [17, Lemma D.1], for any closed geodesic $\ \gamma _m \text{ in } \, ({\Sigma },{ g_{{ poin },m} }) \text{ of } \text{ length } \,{ \ell }_m \le { \ell }_{m,0} = l_{{ g_{{ poin },m} }}(\gamma _m) \le 2 { \ell }_m$, which exists by definition of $\ { \ell }_m$, a neighbourhood $\ U_m \text{ of } \, \gamma _m \text{ in } \,({\Sigma },{ g_{{ poin },m} })\$ isometric to $\ T / \sim$, where

$$\begin{aligned} T := \{ (r e^{i \theta }\ | \ 1 \le r \le e^{{ \ell }_{m,0} (4 \pi (p-1))^{1/2} }, |\theta - \pi /2| < \theta _0\} \end{aligned}$$

is considered in the hyperbolic plane with hyperbolic metric divided by $\ 4 \pi (p-1) > 0\$ in order to adjust to our convention of curvature $\ K_{ g_{{ poin },m} }= - 4 \pi (1-p)$, $\ \sim \$ identifies $\ e^{i \theta } \text{ and } \,e^{{ \ell }_{m,0} + i \theta }$, and $\ \theta _0\$ is a fixed positive constant, as we assume $\ { \ell }_m \le 1$. Here $\ \gamma _m\$ corresponds to $\ \theta = \pi / 2\$, say $\ \gamma _m(t) :\cong e^{t + i \pi / 2}$. We select a second closed geodesic $\ \hat{\gamma }_m(t) :\cong e^{t + i ((\pi + \theta _0)/2)}, 0 \le t \le { \ell }_{m,0} (4 \pi (p-1))^{1/2}$, with length $\ l_{ g_{{ poin },m} }(\hat{\gamma }_m) = { \ell }_{m,0}\$ and which is homotopic to $\ \gamma _m$, hence is non-nullhomotopic, see [17, pp. 184–185].

Since $\ \theta \mapsto e^{i \theta }\$ is a geodesic in the hyperbolic plane and geodesics in the hyperbolic plane are globally miniminzing, we see with hyperbolic distance that

$$\begin{aligned} d_{ g_{{ poin },m} }(\gamma _m(t) , \hat{\gamma }_m(t)) = d_H\left( e^{t + i \pi / 2} , e^{t + i (\pi + \theta _0) / 2}\right) / \sqrt{4 \pi (p-1)} \ge \delta \end{aligned}$$

for some fixed $\ \delta = \delta (\theta _0,p) > 0$, as $\ { \ell }_m \le 1$, in particular

$$\begin{aligned} d_{ g_{{ poin },m} }(\gamma _m , \hat{\gamma }_m) \ge \delta - 2 { \ell }_{m,0} \ge \delta / 2 \quad \text{ for } \,m \text{ large }, \end{aligned}$$

(2.89)

as $\ { \ell }_{m,0} \le 2 { \ell }_m \rightarrow 0$. As both $\ \gamma _m \text{ and } \,\hat{\gamma }_m\$ are non-nullhomotopic, we see by (2.86) and (2.87) that $\ \gamma _m \cap \Omega _{1/4}, \hat{\gamma }_m \cap \Omega _{1/4} \ne \emptyset \text{ for } \,m\$ large and conclude by (2.88) that

$$\begin{aligned} d_{ g_{{ poin },m} }(\gamma _m , \hat{\gamma }_m) \le diam(\Omega _{1/4},{ g_{{ poin },m} }) \rightarrow 0, \end{aligned}$$

which contradicts (2.89), and the proposition is fully proved.

Remark Combining Proposition 2.3 with the previous remark after Proposition 2.2, we see that the conformal structures induced by smooth immersions $\ f: {\Sigma }\rightarrow \mathbb{R }^n\$ of a closed, orientable surface $\ {\Sigma } \text{ of } \text{ genus } \,p \ge 1\$ with

$$\begin{aligned} { \mathcal{W } }(f) \le \min (8 \pi , \tilde{\beta }^n_p) - \delta \end{aligned}$$

lie in a compact subset of moduli space depending on $\ n,p,\delta > 0$. This was already proved in [7, Theorems 5.3 and 5.5] and [14, Theorem 1.1]. $\square$

Now we estimate the conformal factor if no topology is lost under the algebraic energy restriction in order to apply the Hardy space theory. This gives a precise criterion, when the conformal factor can be estimated for a given sequence without applying Möbius transformations.

Proposition 2.4

If in Proposition 2.1 no topology is lost, in the sense that

$$\begin{aligned} genus(spt\ \mu ) = p \ge 1, \end{aligned}$$

(2.90)

and

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < { \mathcal{W } }(\mu ) + { e_{n} }, \end{aligned}$$

(2.91)

then the induced metrics $\ g_m := { g_{euc} }| {\Sigma }_m\$ are uniformly conformal to unit volume constant curvature metrics $\ { g_{{ poin },m} }:= e^{-2 u_m} g_m\$ for $\ m\$ large, more precisely

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } \parallel u_m \parallel _{L^\infty (\Sigma )} < \infty . \end{aligned}$$

(2.92)

Proof

We continue with the notation of the Proposition 2.1 and want to exclude the real bad points in (2.20). Firstly by (2.90), we get $\ genus({\Sigma }) = genus(\tilde{\Sigma }_m) = p = genus({\Sigma }_m)\$ and from (2.23) that

$$\begin{aligned} p_k = 0 \quad \text{ for } \quad k = 1, \ldots , K. \end{aligned}$$

(2.93)

Next as $\ f\$ is a uniformly conformal $\ W^{2,2}$-immersion, we get from [10] Proposition 5.2, Theorem 7.2 and (2.5)

$$\begin{aligned} \displaystyle {\frac{1}{4}} \int \limits _{\Sigma }|A_f|^2 { \ \mathrm{{d}} }\mu _f + 2 \pi (1 - genus({\Sigma })) = { \mathcal{W } }(f) = { \mathcal{W } }(\mu _f) = { \mathcal{W } }(\mu ). \end{aligned}$$

Next by (2.1) and (2.91), we may assume

$$\begin{aligned} { \mathcal{W } }({\Sigma }_m) \le \min (8 \pi , { \mathcal{W } }(\mu ) + { e_{n} }) - \delta \end{aligned}$$

for some $\ \delta > 0 \text{ and } \, m\$ large, and get by the Gauß equations and the Gauß–Bonnet Theorem as in (2.11) and $\ genus({\Sigma }) = genus({\Sigma }_m) = p\$ that

$$\begin{aligned} \int \limits _{{\Sigma }_m} |A_{{\Sigma }_m}|^2 { \ \mathrm{{d}} }{ \mathcal{H }^2 }\le \int \limits _{\mathbb{R }^2} |A_f|^2 { \ \mathrm{{d}} }\mu _f + 4 { e_{n} } - 4 \delta \end{aligned}$$

for $\ m\$ large. Putting $\ \alpha _f := |A_f|^2 \mu _f\$ and recalling $\ \alpha _m := |A_{f_m}|^2 { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \alpha \$ and $\ spt\ \alpha _m \subseteq B_R(0)\$ by (2.2), we get

$$\begin{aligned} \alpha (\mathbb{R }^n) = \lim \limits _{m \rightarrow \infty } \alpha _m(\mathbb{R }^n) \le \alpha _f(\mathbb{R }^n) + 4 { e_{n} } - 4 \delta . \end{aligned}$$

(2.94)

Recalling $\ \tilde{\alpha }_m := |A_{\tilde{f}_m}|^2 { \mathcal{H }^2 }\lfloor \tilde{\Sigma }_m \rightarrow \tilde{\alpha }$, we get by (2.29) for any $\ \eta \in C^0_0(\mathbb{R }^n), \eta \ge 0$, that

$$\begin{aligned} \int \eta { \ \mathrm{{d}} }\alpha _f&= \int \limits _{{\Sigma }} (\eta \circ f) |A_f|^2 { \ \mathrm{{d}} }\mu _g \le \liminf \limits _{m \rightarrow \infty } \int \limits _{{\Sigma }} (\eta \circ \tilde{f}_m) |A_{\tilde{f}_m}|^2 { \ \mathrm{{d}} }\mu _{\tilde{g}_m} \nonumber \\&= \lim \limits _{m \rightarrow \infty } \int \eta { \ \mathrm{{d}} }\tilde{\alpha }_m = \int \eta { \ \mathrm{{d}} }\tilde{\alpha }, \end{aligned}$$

hence

$$\begin{aligned} \alpha _f \le \tilde{\alpha }. \end{aligned}$$

(2.95)

By (2.22), we see that

$$\begin{aligned} \alpha _m = \tilde{\alpha }_m \quad \text{ in } \,\mathbb{R }^n - \cup _{k=1}^N \overline{B_{M r_m}(z_k)} \end{aligned}$$

and as $\ r_m \rightarrow 0\$ and with (2.95)

$$\begin{aligned} \alpha = \tilde{\alpha }\ge \alpha _f \quad \text{ in } \,\mathbb{R }^n - \{ z_1, \ldots , z_N \}. \end{aligned}$$

Since $\ \alpha _f(\{x\}) = 0 \text{ for } \text{ any } \,x \in \mathbb{R }^n$, we get from (2.94)

$$\begin{aligned} \alpha (\mathbb{R }^n)&\le \alpha _f(\mathbb{R }^n) + 4 { e_{n} } - 4 \delta = \alpha _f(\mathbb{R }^n - \{ z_1, \ldots , z_N \}) + 4 { e_{n} } - 4 \delta \nonumber \\&= \alpha (\mathbb{R }^n - \{ z_1, \ldots , z_N \}) + 4 { e_{n} } - 4 \delta \end{aligned}$$

and

$$\begin{aligned} \alpha (\{ z_1, \ldots , z_N \}) \le 4 { e_{n} } - 4 \delta . \end{aligned}$$

(2.96)

Combining with (2.93), we see that (2.20) is vacant for $\ \delta \$ as above. This yields by (2.22) that $\ {\Sigma }_m = \tilde{\Sigma }_m \cong {\Sigma }$, hence $\ \tilde{g}_m = \tilde{f}_m^* g_m, { g_{{ poin },m} }= \tilde{f}_m^* { \tilde{g}_{{ poin },m} }, u_m = \tilde{u}_m \circ f_m$, and (2.92) follows from (2.28), and the proposition is proved. $\square$

3 Compactness in moduli space

In [7], it was proved that compactness in moduli space gives uniformly conformal weak limits in $\ W^{2,2}_{loc}({\Sigma }- \mathcal{{S}}) \text{ for } \text{ some } \text{ finite } \,\mathcal{{S}} \subseteq {\Sigma }$. In our set up, we clarify that if some topology is kept under compactness in moduli space then no additional choice of Möbius transformations is necessary to keep all topology.

Proposition 3.1

If in Proposition 2.1 some topology is kept in the sense that

$$\begin{aligned} genus(spt\ \mu ) \ge 1, \end{aligned}$$

(3.1)

and the conformal structures induced by $\ {\Sigma }_m\ \$ lie in a compact subset of the moduli space, then

$$\begin{aligned} genus(spt\ \mu ) = p, \end{aligned}$$

(3.2)

that is no topology is lost, and

$$\begin{aligned} { [{{\Sigma }_m}] } \rightarrow { [{spt\ \mu }] } \text{ in } \text{ moduli } \text{ space } \end{aligned}$$

(3.3)

in the sense of (2.49).

Proof

We use the notation of Proposition 2.1 and consider a closed, orientable surface $\ {\Sigma }_p \text{ of } \text{ genus } \,p \ge 1\$ and diffeomorphisms $\ f_m: {\Sigma }_p { \stackrel{\approx }{\longrightarrow } }{\Sigma }_m$. As the conformal structures induced by the pull-back metrics $\ g_m := f_m^* { g_{euc} }\$ lie in a compact subset of the moduli space choosing the parametrizations $\ f_m\$ appropriately and passing to a subsequnce, we may assume that the unit volume constant curvature metrics

$$\begin{aligned} { g_{{ poin },m} }:= e^{-2 u_m} g_m \rightarrow { g_{{ poin },0} }\quad \text{ smoothly } \text{ on } {\Sigma }_p. \end{aligned}$$

(3.4)

By (2.1), we may assume

$$\begin{aligned} { \mathcal{W } }(f_m) \le 8 \pi - \delta \end{aligned}$$

(3.5)

for some $\ \delta > 0$. As $\ f_m({\Sigma }_p) = {\Sigma }_m \subseteq B_R(0)\$ by (2.2), we get by the Li-Yau inequality in [11] or [8, A.16]

$$\begin{aligned} \mu _{g_m}({\Sigma }_p) \le C. \end{aligned}$$

(3.6)

We define $\ \nu _m := |A_{f_m}|^2 \mu _{g_m}\$ and see by the Gauß equations and the Gauß–Bonnet theorem as in (2.11) that

$$\begin{aligned} \nu _m({\Sigma }_p) = \int \limits _{{\Sigma }_p} |A_{f_m}|^2 { \ \mathrm{{d}} }\mu _{g_m} = 4 { \mathcal{W } }(f_m) + 8 \pi (p - 1) \le 8 \pi (p + 3) =: A_0 < \infty , \end{aligned}$$

hence after passing to a subsequence $\ \nu _m \rightarrow \nu \$ weakly as Radon measures on $\ {\Sigma }_p$. Clearly $\ \nu ({\Sigma }_p) \le A_0 < \infty$, and there are at most finitely many bad points $\ q_1, \ldots , q_L \in {\Sigma }_p\$ with

$$\begin{aligned} \nu (\{q_l\}) \ge \varepsilon _0(n) \quad \text{ for } \quad l = 1, \ldots , L, \end{aligned}$$

(3.7)

for $\varepsilon _0(n)$ small enough choosen below, and we consider the open set $\Omega _0 := {\Sigma }_p - \{ q_1, \ldots , q_L\}$.

For any $\ q \in \Omega _0\$ there exists an open neighbourhood $\ \varphi : U(q) \cong B_1(0) \text{ of } \, q\$ with

$$\begin{aligned} \nu (\overline{U(q)}) < \varepsilon _0(n) \end{aligned}$$

(3.8)

and for $\ m\$ large that

$$\begin{aligned} \int \limits _{U(q)} |A_{f_m}|^2 { \ \mathrm{{d}} }\mu _{g_m} \le \varepsilon _0(n) - \delta \end{aligned}$$

(3.9)

for some $\ \delta = \delta (q) > 0$. By elementary differential geometry and the Gauß–Bonnet theorem, we know

$$\begin{aligned} - \Delta _{g_m} u_m + 2 \pi \chi ({\Sigma }_p) e^{-2u_m} = K_{g_m} \quad \text{ on } \,{\Sigma }_p. \end{aligned}$$

(3.10)

By the uniformisation theorem for simply connected Riemann surfaces, see [4, Theorem IV.1.1], we can parametrize $\ f_m \circ \varphi _m^{-1}: B_1(0) \cong U(q) \rightarrow \mathbb{R }^n\$ conformally with respect to the euclidean metric on $\ B_1(0)$, possibly after replacing $\ U(q)\$ by a slightly smaller ball. Then by [12] Theorem 4.2.1 and (3.9) for $\ \varepsilon _0(n)\$ small enough, there exists $\ v_m \in C^\infty (U(q))\$ with

$$\begin{aligned} - \Delta _{g_m} v_m = K_{g_m} \quad \text{ on } \,U(q) \end{aligned}$$

(3.11)

satisfying

$$\begin{aligned} \Vert v_m \Vert _{L^\infty (U(q))} \le C(n,\delta ) \int \limits _{B_1(0)} |A_{f_m}|^2 { \ \mathrm{{d}} }\mu _{g_m} \le C(n,q). \end{aligned}$$

(3.12)

Actually one can choose $\varepsilon _0(n) = 4 \pi$, see Proposition 5.1. By (3.10), we see

$$\begin{aligned} - \Delta _{{ g_{{ poin },m} }}(u_m - v_m) = - 2 \pi \chi ({\Sigma }_p) \quad \text{ in } \,U(q), \end{aligned}$$

(3.13)

hence by local maximum estimates, see [5, Theorem 8.17], and (3.4), writing $\ \varphi : U_\varrho (q) \cong B_\varrho (0)\$ that

$$\begin{aligned} \sup \limits _{U_{1/2}(q)} (u_m-v_m) \le C({ g_{{ poin },0} },q) \left( \parallel 2 \pi \chi ({\Sigma }_p) \parallel _{L^2(U_{3/4}(q))} + \parallel (u_m-v_m)_+ \parallel _{L^2(U_{3/4}(q))} \right) \end{aligned}$$

and by (3.12)

$$\begin{aligned} \sup \limits _{U_{1/2}(q)} u_m \le C(n,p,{ g_{{ poin },0} },q) \left( 1 + \parallel u_{m,+} \parallel _{L^2(U_{3/4}(q))} \right) \end{aligned}$$

for $\ m\$ large. To estimate the norm on the right-side, we observe by (3.6)

$$\begin{aligned} C \ge \mu _{g_m}({\Sigma }_p) \ge \int \limits _{U_{3/4}(q)} e^{2 u_m} { \ \mathrm{{d}} }\mu _{ g_{{ poin },m} }\ge \int \limits _{U_{3/4}(q)} (1 + u_m)_+^2 { \ \mathrm{{d}} }\mu _{ g_{{ poin },m} }\end{aligned}$$

and get for $\ m\$ large

$$\begin{aligned} \sup \limits _{U_{1/2}(q)} u_m, |D f_m| \le C(n,p,{ g_{{ poin },0} },q). \end{aligned}$$

(3.14)

Then $\ \Gamma - u_m + v_m \ge 0 \text{ in } \,U_{1/2}(q) \text{ for } \,\Gamma = C(n,p,{ g_{{ poin },0} },q)$, hence by (3.13) and the Harnack inequality, see [5] Theorem 8.17 and 8.18, and (3.4) that

$$\begin{aligned} \sup \limits _{U_{1/4}(q)} (\Gamma - u_m - v_m) \le C({ g_{{ poin },0} },q) \left( \inf \limits _{U_{1/4}(q)} (\Gamma - u_m - v_m)\,+\parallel 2 \pi \chi ({\Sigma }_p) \parallel _{L^2(U_{1/2}(q))} \right) \end{aligned}$$

and by (3.12)

$$\begin{aligned} \sup \limits _{U_{1/4}(q)} (\Gamma - u_m) \le C(n,p,{ g_{{ poin },0} },q) \inf \limits _{U_{1/4}(q)} (\Gamma - u_m) + C(n,p,{ g_{{ poin },0} },q), \end{aligned}$$

(3.15)

Now if $\ \liminf _{m \rightarrow \infty } \inf _{U_{1/4}(q)} u_m > - \infty$, we see from (3.14) that $\ u_m\$ is bounded from below and above on $\ U_{1/4}(q)$. Otherwise for a subsequence $\ \inf _{U_{1/4}(q)} u_m \rightarrow - \infty$. Then

$$\begin{aligned} \sup \limits _{U_{1/4}(q)} (\Gamma - u_m) = \Gamma - \inf \limits _{U_{1/4}(q)} u_m \rightarrow \infty \end{aligned}$$

and by (3.15)

$$\begin{aligned} \infty \leftarrow \inf \limits _{U_{1/4}(q)} (\Gamma - u_m) = \Gamma - \sup \limits _{U_{1/4}(q)} u_m, \end{aligned}$$

hence $u_m \rightarrow -\infty \text{ uniformly } \text{ on } \,U_{1/4}(q)$.

Covering appropriately, we get after passing to a subsequence either

$$\begin{aligned} \begin{array}{ll} &{} u_m \rightarrow -\infty \text{ uniformly } \text{ on } \text{ compact } \text{ subsets } \text{ of } \,\Omega _0, \\ \text{ or } \\ &{} u_m \text{ is } \text{ uniformly } \text{ bounded } \text{ on } \text{ compact } \text{ subsets } \text{ of } \, \Omega _0. \\ \end{array} \end{aligned}$$

(3.16)

We choose open neighbourhoods $\ U(q_l) \text{ of } \,q_l\$ which are pairwise disjoint discs and put

$$\begin{aligned} \Omega ^{\prime } := {\Sigma }_p - \cup _{l=1}^L U(q_l) \subset \subset \Omega _0. \end{aligned}$$

(3.17)

In the first case in (3.16), we see by (3.4) that $\ diam_{g_m}(\Omega ^{\prime }) \rightarrow 0\$ for the intrinsic diameter, hence

$$\begin{aligned} diam(f_m(\Omega ^{\prime })) \rightarrow 0. \end{aligned}$$

In the notation of Proposition 2.1, as $\ {\Sigma }_m = f_m({\Sigma }_p) \subseteq \cup _{k=1}^K B_{\varrho _k/2}(x_k) \text{ for } \,m\$ large, we get after passing to a subsequence

$$\begin{aligned} f_m(\Omega ^{\prime }) \subseteq B_{5 \varrho _k / 8}(x_k) \cap {\Sigma }_m \subseteq D^k_m = {\Sigma }_m \cap B_\sigma (x_k) \quad \text{ for } \text{ some } \,k \in \{ 1, \ldots , K \} \end{aligned}$$

(3.18)

for some appropriate $\ \sigma \in [5 \varrho _k / 8 , 7 \varrho _k / 8]\$ and $\ m\$ large. We know that $\ D^k_m\$ is connected, and its boundary $\ \partial D^k_m\$ consists of a single Jordan curve. Therefore the complement of $\ {\Sigma }_m - \partial D^k_m = {\Sigma }_m - \partial B_\sigma (x_k)\$ consists of exactly two components which are

$$\begin{aligned} D^k_m \quad \text{ and } \quad {\Sigma }_m - \overline{B_\sigma (x_k)}. \end{aligned}$$

As $\ f_m\$ is a diffeomorphism, we know that $\ \gamma _m := f_m^{-1}(\partial D^k_m)\$ is a single Jordan curve in ${\Sigma }_p\$, in particular connected, and, as $\ \gamma _m \cap \Omega ^{\prime } = \emptyset \$ by (3.18), we conclude

$$\begin{aligned} \gamma _m \subseteq U(q_l) \quad \text{ for } \text{ some } \,l \in \{ 1, \ldots , L \} \end{aligned}$$

after passing to a subsequence. As $\ U(q_l)\$ is a disc, the complement of $\ \gamma _m \text{ in } \,{\Sigma }_p\$ consists of exactly two components, one of which is the interior $\ I_m \text{ of } \, \gamma _m \text{ in } \,U(q_l)\$ which moreover is a disc as well. We call the other component the exterior $\ E_m$, and see that $\ E_m\$ is not a disc, as $\ I_m \oplus E_m = {\Sigma }_p \not \cong S^2$. These components correspond under $\ f_m\$ to the components of $\ {\Sigma }_m - \partial D^k_m$.

By (3.18), we see $\ f_m^{-1}(D^k_m) \not \subseteq U(q_l)$, hence $\ D^k_m \cong f_m^{-1}(D^k_m) = E_m\$ is not a disc and

$$\begin{aligned} {\Sigma }_m - \overline{B_\sigma (x_k)} = f_m^{-1}(I_m) \text{ is } \text{ a } \text{ disc }. \end{aligned}$$

Then $\ D^k_m\$ appears in (2.20) and is replaced in the construction of Proposition 2.1 by a disc, and we get

$$\begin{aligned} genus({\Sigma }) = genus(\tilde{\Sigma }_m) \le genus \left( ({\Sigma }_m - \overline{B_\sigma (x_k)} ) \oplus B_1(0) \right) = 0. \end{aligned}$$

But $\ genus({\Sigma }) = genus(spt\ \mu ) \ge 1\$ by (3.1), hence can exclude the first case in (3.16).

Therefore we have the second case in (3.16) and prove that no topology is lost in Proposition 2.1 in the sense of (3.2). If (3.2) is not satisfied, we get from (2.23) that $\ \tilde{N} \ge 1\$ and after renumbering in (2.20) that

$$\begin{aligned} p_1 \ge 1. \end{aligned}$$

(3.19)

This means that $\ D^1_m\$ is not a disc and by construction in (2.22), there exists a closed curve $\ \gamma _m \text{ in } \,B_{M r_m}(x_1) \cap {\Sigma }_m\$ which is not null-homotopic in $\ {\Sigma }_m$. As $\ \gamma _m^{{\Sigma }_p} := f_m^{-1}(\gamma _m)\$ is connected and $\ U(q_l)\$ are discs, it has to meet $\ \Omega ^{\prime }\$ in (3.17), hence

$$\begin{aligned} q_m \in \gamma _m^{{\Sigma }_p} \cap \Omega ^{\prime } \ne \emptyset . \end{aligned}$$

Passing to a subsequence, we get $\ q_m \rightarrow q \in \overline{\Omega ^{\prime }} \subseteq \Omega _0$, and we select an open neighbourhood $\ U(q) \subset \subset \Omega _0 \text{ of } \,q\$ which is a disc, say $\ \varphi : U(q) { \stackrel{\approx }{\longrightarrow } }B_1(0), \varphi : U_\varrho (q) \cong B_\varrho (0)$. Clearly for $\ m\$ large, we have $\ q_m \in U(q) \cap \gamma _m^{{\Sigma }_p}$. As $\ U(q)\$ is a disc and $\ \gamma _m^{{\Sigma }_p}\$ is not null-homotopic, $\ \gamma _m^{{\Sigma }_p}\$ cannot stay in $\ U(q)$, hence there is

$$\begin{aligned} q_{m,\varrho } \in \gamma _m^{{\Sigma }_p} \cap \partial U_\varrho (q) \quad \text{ for } \quad 0 < \varrho < 1. \end{aligned}$$

(3.20)

As $\ f_m(\gamma _m^{{\Sigma }_p}) = \gamma _m \subseteq B_{M r_m}(x_1)$, we get

$$\begin{aligned} |f_m(q_{m,\varrho }) - x_1| \rightarrow 0 \quad \text{ for } \quad 0 < \varrho < 1. \end{aligned}$$

(3.21)

Next choosing

$$\begin{aligned} \Omega ^{\prime } \cup U(q) \subset \subset \Omega ^{\prime \prime } \subset \subset \Omega ^{\prime \prime \prime } \subset \subset \Omega _0 \end{aligned}$$

and writing

$$\begin{aligned} \Delta _{ g_{{ poin },m} }f_m = e^{2 u_m} \Delta _{g_m} f_m = e^{2 u_m} \overrightarrow{\mathbf{H }}_{f_m} \quad \text{ on } \,{\Sigma }_p, \end{aligned}$$

we estimate by (3.16)

$$\begin{aligned} \int \limits _{\Omega ^{\prime \prime \prime }} |e^{2 u_m} \overrightarrow{\mathbf{H }}_{f_m}|^2 { \ \mathrm{{d}} }\mu _{ g_{{ poin },m} }\le C \int \limits _{{\Sigma }_p} |\overrightarrow{\mathbf{H }}_{f_m}|^2 { \ \mathrm{{d}} }\mu _{g_m} \le C \end{aligned}$$

and get by standard elliptic theory, see [5, Theorem 8.8], (2.2) and (3.4) for $\ m\$ large

$$\begin{aligned} \parallel f_m \parallel _{W^{2,2}(\Omega ^{\prime \prime })}\le C, \end{aligned}$$

hence after passing to a subsequence using (3.4)

$$\begin{aligned} \begin{array}{c} f_m \rightarrow f_0 \text{ weakly } \text{ in } \,W^{2,2}_{loc}(\Omega _0), \text{ weakly }^* \text{ in } W^{1,\infty }_{loc}(\Omega _0), \\ g_m \rightarrow g_0 := f_0^* { g_{euc} } \text{ weakly } \text{ in } \,W^{1,2}_{loc}(\Omega _0), \\ u_m \rightarrow u_0 \text{ weakly } \text{ in } \,W^{1,2}_{loc}(\Omega _0), \text{ weakly }^* \text{ in } L^\infty _{loc}(\Omega _0), \\ g_0 = f_0^* { g_{euc} }= e^{2 u_0} { g_{{ poin },0} }, \\ \end{array} \end{aligned}$$

(3.22)

with $\ u_0 \in L^\infty _{loc}(\Omega _0)\$ and in particular, we have $\ f_m \rightarrow f_0\$ uniformly on compact subsets of $\ \Omega _0$. Passing to a subsequence in (3.20) and (3.21) for fixed $\ 0 < \varrho < 1$, we get $\ q_{m,\varrho } \rightarrow q_\varrho \$ for this subsequence and

$$\begin{aligned} q_\varrho \in \partial U_\varrho (q), \quad f_0(q_\varrho ) = x_1, \end{aligned}$$

(3.23)

in particular $\ q_\varrho \ne q_{\tilde{\varrho }} \text{ for } \, \varrho \ne \tilde{\varrho }$.

Now for any $\ \eta \in C^0(\mathbb{R }^n), \eta \ge 0,\$ and Fatou’s lemma by (3.22)

$$\begin{aligned} \int \limits _{\Omega _0} (\eta \circ f_0) { \ \mathrm{{d}} }\mu _{g_0} \le \liminf \limits _{m \rightarrow \infty } \int \limits _{\Omega _0} (\eta \circ f_m) { \ \mathrm{{d}} }\mu _{g_m} = \liminf \limits _{m \rightarrow \infty } \int \eta { \ \mathrm{{d}} }\mu _{f_m} \end{aligned}$$

hence for $\ \mu _{f_0} := f_0(\mu _{g_0} | \Omega _0)\$ and observing $\ \mu _{f_m} = { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \$ by (2.2)

$$\begin{aligned} \int \limits _{\Omega _0} \eta { \ \mathrm{{d}} }\mu _{f_0} \le \int \eta { \ \mathrm{{d}} }\mu , \end{aligned}$$

and

$$\begin{aligned} \mu _{f_0} \le \mu . \end{aligned}$$

(3.24)

As $\ u_0 \in L^\infty _{loc}(\Omega _0)$, we get by Proposition 7.1 and (2.7) that

$$\begin{aligned} \# f_0^{-1}(x) \le \theta ^2(\mu _{f_0},x) \le \theta ^2(\mu ,x) < 2 \quad \text{ for } \text{ all } \quad x \in \mathbb{R }^n, \end{aligned}$$

hence

$$\begin{aligned} f_0: \Omega _0 \rightarrow \mathbb{R }^n \text{ is } \text{ injective }. \end{aligned}$$

(3.25)

As $\ \{ q_\varrho | 0 < \varrho < 1 \} \subseteq f_0^{-1}(x_1) \cap \Omega ^{\prime \prime }$, this is a contradiction, and we get (3.2) and may choose $\ {\Sigma }= {\Sigma }_p$.

It remains to prove (3.3). We know by (3.4) that

$$\begin{aligned} { [{{\Sigma }_m}] } = { [{{ g_{{ poin },m} }}] } \rightarrow { [{{ g_{{ poin },0} }}] }, \end{aligned}$$

and, as $\ { [{spt\ \mu }] } = { [{f^* { g_{euc} }}] } = { [{{ g_{{ poin }} }}] }\$ by (2.29) and (2.49), we have to prove

$$\begin{aligned} { [{{ g_{{ poin },0} }}] } = { [{{ g_{{ poin }} }}] }. \end{aligned}$$

(3.26)

From Proposition 7.1, we know that $\ \theta ^2(\mu _{f_0}) > 0 \text{ on } \,f_0(\Omega _0)$, hence $\ f_0(\Omega _0) \subseteq spt\ \mu \$ by (3.24). Since $\ f: {\Sigma }\, { \stackrel{\approx }{\longrightarrow } }\, spt\ \mu \$ is bi-lipschitz by Proposition 2.1 and $\ f_0\$ is locally lipschitz, as $\ u_0 \in L^\infty _{loc}(\Omega _0)\$ in (3.22), we see that

$$\begin{aligned} \phi : f^{-1} \circ f_0: \Omega _0 \rightarrow {\Sigma }\end{aligned}$$

is locally lipschitz and is injective by (3.25). Clearly $\ f \circ \phi = f_0\$ and by (2.29) and (3.22)

$$\begin{aligned} e^{2u_0} { g_{{ poin },0} }= f_0^* { g_{euc} }= \phi ^* f^* { g_{euc} }= \phi ^*(e^{2u} { g_{{ poin }} }) = e^{2 u \circ \phi } \phi ^* { g_{{ poin }} }, \end{aligned}$$

(3.27)

hence

$$\begin{aligned} \phi : (\Omega _0,{ g_{{ poin },0} }) \rightarrow ({\Sigma },{ g_{{ poin }} }) \end{aligned}$$

is conformal, hence holomorphic after possibly reversing the orientation, in particular smooth. As $\ u \in L^\infty ({\Sigma }), u_0 \in L^\infty _{loc}(\Omega _0)\$ and $\ D \phi \$ is continuous, we see that $\ D \phi \$ has full rank everywhere on $\ \Omega _0$. Then $\ \phi$, being injective, is a diffeomorphism onto the open set $\ \Omega := \phi (\Omega _0) \subseteq {\Sigma }$.

Recalling $\ \Omega _0 = {\Sigma }- \{ q_1, \ldots , q_l \}$, we choose open neighbourhoods $\ U(q_l) \text{ of } \,q_l$, which are pariwise disjoint discs, and smooth closed Jordan curves $\ \gamma ^0_l \text{ in } \,U(q_l)\$ such that $\ q_l\$ lie in the interior $\ I^0_l \text{ of } \,\gamma ^0_l \text{ in } \,U(q_l)$. Then $\ U_0 := \Omega _0 - \cup _{l=1}^L \overline{I^0_l} \subset \subset \Omega _0\$ is open with $\ l\$ boundary components and $\ \chi (U_0) = \chi ({\Sigma }) - l$. As $\ \phi \$ is a diffeomorphism, we see that $\ U := \phi (U_0) \subseteq {\Sigma }\$ is open and

$$\begin{aligned} \chi (U) = \chi ({\Sigma }) - l. \end{aligned}$$

(3.28)

Clearly $\ \phi (\partial U_0) \subseteq \partial U$. On the other hand for $\ w \in \partial U$, there exists $\ y_k \in U_0 \text{ with } \,\phi (y_k) \rightarrow w$. For a subsequence we see $\ y_k \rightarrow y \in \overline{U_0} \subseteq \Omega _0$, hence $\ w = \phi (y) \in \Omega$. If $\ y \in U_0$, then $\ w \in U$, which is a contradiction. Therefore $\ y \in \partial U_0 \text{ and } \,w \in \phi (\partial U_0)$. Together

$$\begin{aligned} \partial U = \phi (\partial U_0) = \cup _{l=1}^L \phi (\gamma ^0_l), \end{aligned}$$

and the boundary of $\ U \text{ in } \,{\Sigma }\$ consists of $\ l\$ Jordan curves $\ \gamma _l := \phi (\gamma ^0_l) \subseteq \Omega$. The boundary of the exterior $\ V := {\Sigma }- \overline{U} \text{ of } \, U\$ lies in $\ \partial U = \cup _{l=1}^L \gamma _l$, hence $\ \partial V\$ consists of at most $\ l\$ Jordan curves. As $\ \chi (V) = \chi ({\Sigma }) - \chi (U) = l\$ by (3.28), we see that each component of $\ V = {\Sigma }- \overline{U}\$ is a disc.

Since $\phi (I^0_l - \{ q_l \})$ is connected and lies in $\ \Omega - \overline{U} \subseteq {\Sigma }- \overline{U} = V$, it is contained in one of these discs, which we call $\ D_l$. By the uniformisation theorem for simply connected Riemann surfaces, see [4, Theorem IV.1.1], these are conformally equivalent to the disc or the plane in $\ \mathbb{C }$. Extending beyond the Jordan curves $\ \gamma ^0_l \text{ and } \,\gamma _l$, we get conformal diffeomorphisms $\ \varphi _l: B_1(0)\, { \stackrel{\approx }{\longrightarrow } }\, (I^0_l,{ g_{{ poin },0} }), \varphi _l(0) = q_l, \psi _l: B_1(0) { \stackrel{\approx }{\longrightarrow } }(D_l,{ g_{{ poin }} })$. Then

$$\begin{aligned} h_l := \psi _l^{-1} \circ \phi \circ \varphi _l: B_1(0) - \{0\} \rightarrow B_1(0) \end{aligned}$$

is holomorphic and injective. Therefore $\ h_l\$ extends to a holomorphic function on $\ B_1(0) \text{ with } \,h^{\prime }_l(0) \ne 0$, and $\ \phi \$ extends to a holomorphic function $\ \phi : ({\Sigma },{ g_{{ poin },0} }) \rightarrow ({\Sigma },{ g_{{ poin }} })\$ and $\ D \phi \$ has full rank everywhere on $\ {\Sigma }$. Then $\ \phi ({\Sigma })\$ is open and compact, hence $\ \phi ({\Sigma }) = {\Sigma }\$ and $\ \phi \$ is surjective. Also $\ \phi \$ is a local diffeomorphism, hence a covering projection, as $\ {\Sigma }\$ is compact. Then $\ \# \phi ^{-1}(q)\$ is finite and constant for $\ q \in {\Sigma }$. As $\ \phi \$ is injective on $\ \Omega _0 = {\Sigma }- \{ q_1, \ldots , q_l \}$, we see that $\ \phi \$ is injective, hence a diffeomorphism. From (3.27), we see $\ \phi ^* { g_{{ poin }} }= e^{2 u_0 - 2 (u \circ \phi )} { g_{{ poin },0} }$, hence $\ \phi ^* { g_{{ poin }} }= { g_{{ poin },0} }$, which yields (3.26), and the proposition is proved. $\square$

Combining the previous proposition with §2, we estimate the conformal factor and get the equivalence of no topologiccal loss after applying appropriate Möbius transformations and compactness in moduli space.

Theorem 3.1

Let $\ {\Sigma }_m \subseteq \mathbb{R }^n\$ be closed, orientable, embedded surfaces of fixed genus $\ p \ge 1\$ with

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < 8 \pi , \end{aligned}$$

(3.29)

$$\begin{aligned}&{\Sigma }_m \subseteq B_1(0), \nonumber \\&{ \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \ne 0 \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures }. \end{aligned}$$

(3.30)

Then $\ spt\ \mu \$ is a closed, orientable, embedded topological surface of $\ genus(spt\ \mu ) \le p$. No topology is lost in the sense that

$$\begin{aligned} genus(spt\ \mu ) = p \end{aligned}$$

(3.31)

if and only if some topology is kept in the sense that

$$\begin{aligned} genus(spt\ \mu ) \ge 1 \end{aligned}$$

(3.32)

and the conformal structues

$$\begin{aligned} { [{{\Sigma }_m}] } \text{ lie } \text{ in } \text{ a } \text{ compact } \text{ subset } \text{ of } \text{ the } \text{ moduli } \text{ space }. \end{aligned}$$

(3.33)

In this case if moreover

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < { \mathcal{W } }(\mu ) + { e_{n} }, \end{aligned}$$

(3.34)

then the induced metrics $\ g_m := { g_{euc} }| {\Sigma }_m\$ are uniformly conformal to unit volume constant curvature metrics $\ { g_{{ poin },m} }:= e^{-2 u_m} g_m\$ for $\ m\$ large, more precisely

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } \parallel u_m \parallel _{L^\infty (\Sigma )} < \infty . \end{aligned}$$

(3.35)

Proof

Proposition 2.1 implies that $\ spt\ \mu \$ is a closed, orientable, embedded topological surface of $\ genus(spt\ \mu ) \le p$. By Proposition 2.4 the assumptions (3.31) and (3.34) imply (3.35). It remains to prove the equivalence of (3.31) on the ond side and (3.32) and the relative compactness of the conformal structures $\ { [{{\Sigma }_m}] }\$ on the other side.

When no topology is lost in the sense of (3.31), the conformal structures lie in a compact subset of moduli space by Proposition 2.3 and obviously we have (3.32).

Conversely if some topology is kept in the sense of (3.32) and if the conformal structures lie in a compact subset of moduli, we get (3.31) by Propsition 3.1. $\square$

Theorem 3.2

Let $\ {\Sigma }_m \subseteq \mathbb{R }^n\$ be closed, orientable, embedded surfaces of genus $\ p \ge 1\$ with

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma }_m) < 8 \pi . \end{aligned}$$

Then the conformal structures induced by $\ {\Sigma }_m\ \$ lie in a compact subset of the moduli space if and only if no topology is lost after applying appropriate Möbius transformations, more precisely that any subsequence has a subsequence such that after applying appropriate Möbius transformations

$$\begin{aligned} { \mathcal{H }^2 }\lfloor {\Sigma }_m \rightarrow \mu \quad \text{ weakly } \text{ as } \text{ Radon } \text{ measures } \end{aligned}$$

with $\ spt\ \mu \$ is a closed, orientable, embedded topological surface and

$$\begin{aligned} genus(spt\ \mu ) = p. \end{aligned}$$

In this case after passing to a subsequence the conformal structures converge

$$\begin{aligned} { [{{\Sigma }_m}] } \rightarrow { [{spt\ \mu }] } \text{ in } \text{ moduli } \text{ space }. \end{aligned}$$

Proof

When no topology is lost for all subsequences, the conformal structures lie in a compact subset of moduli space by Proposition 2.3.

Conversely by Proposition 2.2 any subsequence has a subsequence such that after applying appropriate Möbius transformations some topology is kept in the sense that $\ genus(spt\ \mu ) \ge 1$, and when the conformal structures lie in a compact subset of moduli, we get by Propsition 3.1 that $\ genus(spt\ \mu ) = p\$ and the convergence of the conformal structues as above. $\square$

4 Main results

For a closed, orientable surface $\ {\Sigma }\$ with smooth metric $\ g_0\$ or at least uniformly conformal to a smooth metric, we recall the definition in (1.7)

$$\begin{aligned} { \mathcal{W } }({\Sigma },g_0,n) := \inf \{ { \mathcal{W } }(f)\ | \ f: {\Sigma }\rightarrow \mathbb{R }^n \text{ smooth } \text{ immersion } \text{ conformal } \text{ to } \,g_0\}. \end{aligned}$$

Theorem 4.1

Let $\ f: {\Sigma }\rightarrow \mathbb{R }^n\$ be a smooth immersion of a closed, orientable surface $\ {\Sigma }\not \cong S^2\$ with

$$\begin{aligned} { \mathcal{W } }(f) \le \min (8 \pi , { \mathcal{W } }({\Sigma },f^* { g_{euc} },n) + { e_{n} } ) - \delta \end{aligned}$$

(4.1)

for some $\ \delta > 0$. Moreover we assume that the conformal structure induced by the pull-back metric of $\ f$ lies in a compact subset $\ K$ of the moduli space.

Then after applying an Möbius transformation, the pull-back metric $\ g := f^* { g_{euc} }\$ is uniformly conformal to a unit volume constant curvature metric $\ { g_{{ poin }} }:= e^{-2 u} g$, more precisely

$$\begin{aligned} \parallel u \parallel _{L^\infty ({\Sigma })} \le C(n,p,K,\delta ). \end{aligned}$$

Proof

We consider a sequence of smooth immersions $\ f_m: {\Sigma }\rightarrow \mathbb{R }^n\$ with pull-back metrics $\ g_m := f_m^* { g_{euc} }$,

$$\begin{aligned} { \mathcal{W } }(f_m) \le \min (8 \pi , { \mathcal{W } }({\Sigma },g_m,n) + { e_{n} } ) - \delta \end{aligned}$$

(4.2)

and with conformal structures induced by $\ g_m\$ converging in moduli space. As $\ { \mathcal{W } }(f_m) \le 8 \pi - \delta \$ by (4.2), we apply Proposition 2.2 and can proceed after applying appropriate Möbius transformations and passing to a subsequence as in Proposition 2.1 with some topology kept, that is $\ genus(spt\ \mu ) \ge 1$. By Proposition 3.1 no topology is lost in the sense that $\ genus(spt\ \mu ) = genus({\Sigma })$, and we get a uniformly conformal $\ W^{2,2}-$immersion $\ f: {\Sigma }\rightarrow \mathbb{R }^n\$ with

$$\begin{aligned} { [{g_m}] } = { [{f_m({\Sigma })}] } \rightarrow { [{spt\ \mu }] } = { [{f^* { g_{euc} }}] } \text{ in } \text{ moduli } \text{ space } \end{aligned}$$

by (3.3). Then by [10, Proposition 5.1 and Theorem 5.1]

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma },g_m,n) \le { \mathcal{W } }({\Sigma },f^* { g_{euc} },n) \le { \mathcal{W } }(f), \end{aligned}$$

hence by (4.2)

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } { \mathcal{W } }(f_m) < { \mathcal{W } }(f) + { e_{n} }, \end{aligned}$$

which verifies (2.91). Then by Proposition 2.4

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } \parallel u_m \parallel _{L^\infty ({\Sigma })} < \infty , \end{aligned}$$

and the theorem follows, as $\ f_m\$ was an arbitrary sequence. $\square$

As $\ { \mathcal{W } }({\Sigma },g,n) \ge \beta _p^n \text{ and } \,{ e_{3} } = 4 \pi$, we obtain the following corollary, which may be considered as a pratial converse of [9, Lemma 5.1].

Theorem 4.2

Let $\ f: {\Sigma }\rightarrow \mathbb{R }^n\$ be a smooth immersion of a closed, orientable surface $\ {\Sigma } \text{ of } \text{ genus } \,p \ge 1\$ with

$$\begin{aligned} { \mathcal{W } }(f) \le \left\{ \begin{array}{ll} 8 \pi - \delta &{} \quad \text{ for } \ n = 3, \\ \beta _p^n + { e_{n} } - \delta &{} \quad \text{ for } \ n \ge 4, \\ \end{array} \right. \end{aligned}$$

(4.3)

for some $\ \delta > 0\$ and assume that the conformal structure induced by the pull-back metric of $\ f\$ lies in a compact subset $\ K\$ of the moduli space.

Then after applying an Möbius transformation, the pull-back metric $\ g := f^* { g_{euc} }\$ is uniformly conformal to a unit volume constant curvature metric $\ { g_{{ poin }} }:= e^{-2 u} g$, more precisely

$$\begin{aligned} \parallel u \parallel _{L^\infty ({\Sigma })} \le C(n,p,K,\delta ). \end{aligned}$$

(4.4)

$\square$

Also we generalize the lower semicontinuity of $\ { \mathcal{W } }({\Sigma },g,n) \text{ with } \text{ respect } \text{ to } g$ of [10] Proposition 5.1 below the energy level $\ { \mathcal{W } }_{n,p}\$ to the energy level $\ 8 \pi$. We write $\ { \mathcal{W } }({\Sigma },c,n) := { \mathcal{W } }({\Sigma },g,n)\$ for the conformal structure $\ c \text{ induced } \text{ by } \,g$.

Proposition 4.1

Let ${\Sigma }\not \cong S^{2}$ be a closed, orientable surface and consider conformal structures $c_m \rightarrow c$ converging in moduli space. Then

$$\begin{aligned} \liminf \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma },c_m,n) < 8 \pi \Longrightarrow { \mathcal{W } }({\Sigma },c,n) \le \liminf \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma },c_m,n). \end{aligned}$$

In particular $c \mapsto { \mathcal{W } }({\Sigma },c,n) \text{ is } \text{ continuous } \text{ at } \, c \text{ with } { \mathcal{W } }({\Sigma },c,n) \le 8 \pi$.

Proof

We select unit volume constant curvature metrics $\ { g_{{ poin },m} }, { g_{{ poin }} } \text{ inducing } \,c_m,c\$ with

$$\begin{aligned} { g_{{ poin },m} }\rightarrow { g_{{ poin }} }\quad \text{ smoothly } \text{ on } \,{\Sigma }\end{aligned}$$

and smooth immersion $\ f_m: {\Sigma }\rightarrow \mathbb{R }^n\$ conformal to $\ { g_{{ poin },m} }\$ and with

$$\begin{aligned} { \mathcal{W } }(f_m) \le { \mathcal{W } }({\Sigma },c_m,n) + 1/m. \end{aligned}$$

Clearly by assumption and $\ { e_{n} } \ge 2 \pi$, we see

$$\begin{aligned} { \mathcal{W } }(f_m) \le \min (8 \pi , { \mathcal{W } }({\Sigma },c_m,n) + { e_{n} }) - \delta \end{aligned}$$

for $\ \delta > 0\$ small enough, $\ m\$ large, and get by Theorem 4.1 after applying appropriate Möbius transformations for the pull-back metrics $\ g_m := f_m^* { g_{euc} }= e^{2 u_m} { g_{{ poin },m} }\$ that

$$\begin{aligned} \limsup \limits _{m \rightarrow \infty } \parallel u_m \parallel _{L^\infty ({\Sigma })} < \infty . \end{aligned}$$

Then by Proposition 6.1 and the remark following after passing to an appropriate subsequence

$$\begin{aligned}&f_m \rightarrow f \text{ weakly } \text{ in } \,W^{2,2}({\Sigma }), \nonumber \\&f^* { g_{euc} }= e^{2u} { g_{{ poin }} }, \end{aligned}$$

for some $\ u \in L^\infty ({\Sigma })$. Therefore $\ f \text{ is } \text{ a } \, W^{2,2}-$immersion uniformly conformal to $\ { g_{{ poin }} }$, and we get from [10, Theorem 5.1]

$$\begin{aligned} { \mathcal{W } }({\Sigma },c,n) \le { \mathcal{W } }(f) \le \liminf \limits _{m \rightarrow \infty } { \mathcal{W } }(f_m) = \liminf \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma },c_m,n). \end{aligned}$$

Finally, if $\ c \mapsto { \mathcal{W } }({\Sigma },c,n)\$ were not continuous at $\ c \text{ with } \, { \mathcal{W } }({\Sigma },c,n) \le 8 \pi$, by upper semicontinuity of $\ c \mapsto { \mathcal{W } }({\Sigma },c,n)\$ in [10, Proposition 5.1], there exists a sequence $\ c_m \rightarrow c\$ with

$$\begin{aligned} \liminf \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma },c_m,n) < { \mathcal{W } }({\Sigma },c,n) \le 8 \pi . \end{aligned}$$

Then by above

$$\begin{aligned} { \mathcal{W } }({\Sigma },c,n) \le \liminf \limits _{m \rightarrow \infty } { \mathcal{W } }({\Sigma },c_m,n), \end{aligned}$$

which is a contradiction, and the proposition is proved. $\square$

Appendices

Appendix A: Estimates in higher dimension

In this section, we prove a higher dimensional version of [9, Theorem 6.1]. We recall the definition of the constants in (1.2)

$$\begin{aligned} { e_{n} } := \left\{ \begin{array}{l@{\quad }l@{\quad }l} 4 \pi &{} \text{ for }&{}n = 3, \\ 8 \pi / 3 &{} \text{ for }&{} n = 4, \\ 2 \pi &{} \text{ for } &{} n \ge 5. \\ \end{array} \right. \end{aligned}$$

Theorem 5.1

Let $\ f: \mathbb{R }^2 \rightarrow \mathbb{R }^n\$ be a complete conformal immersion with induced metric $\ g = e^{2u} { g_{euc} }\$ and square integrable second fundamental form satisfying

$$\begin{aligned} \int \limits _{\mathbb{R }^2} K_g { \ \mathrm{{d}} }\mu _g = 0, \end{aligned}$$

(5.1)

$$\begin{aligned} \int \limits _{\mathbb{R }^2} |A|^2 { \ \mathrm{{d}} }\mu _g \le 4 { e_{n} } - \delta := \left\{ \begin{array}{l@{\quad }l@{\quad }l} 16 \pi - \delta &{} \text{ for }&{} n = 3, \\ 32 \pi / 3 - \delta &{} \text{ for }&{} n = 4, \\ 8 \pi - \delta &{} \text{ for }&{} n \ge 5, \\ \end{array} \right. \end{aligned}$$

(5.2)

for some $\ \delta > 0$. Then the limit $\lambda = \lim _{z \rightarrow \infty } u(z) \in \mathbb{R }$ exists, and

$$\begin{aligned} \Vert u - \lambda \Vert _{L^\infty (\mathbb{R }^2)}, \Vert D u\Vert _{L^2(\mathbb{R }^2)}, \Vert D^2 u\Vert _{L^1(\mathbb{R }^2)} \le C(n,\delta ) \int \limits _{\mathbb{R }^2} |A|^2 { \ \mathrm{{d}} }\mu _g. \end{aligned}$$

(5.3)

Proof

For $\ n = 3$, we estimate with $\ |K| \le |A|^2/2\$ that

$$\begin{aligned} \int \limits _{\mathbb{R }^2} |K| { \ \mathrm{{d}} }\mu _g \le \frac{1}{2} \int \limits _{\mathbb{R }^2} |A|^2 { \ \mathrm{{d}} }\mu _g \le 8 \pi - \delta /2, \end{aligned}$$

and the result follows from [9, Theorem 6.1]. For $\ n = 4$, we estimate with $\ |A^0|^2 = |A|^2/2 - K\$ and (5.1) that

$$\begin{aligned} \int \limits _{\mathbb{R }^2} |K| { \ \mathrm{{d}} }\mu _g + \frac{1}{2} \int \limits _{\mathbb{R }^2} |A^0|^2 { \ \mathrm{{d}} }\mu _g \le \frac{3}{4} \int \limits _{\mathbb{R }^2} |A|^2 { \ \mathrm{{d}} }\mu _g \le 8 \pi - 3 \delta /4, \end{aligned}$$

and again the result follows from [9, Theorem 6.1].

For $\ n \ge 5$, we get by elementary differential geometry from $\ g_{euc} = e^{-2u} g \text{ and } \,K_{g_{euc}} = 0\$ that

$$\begin{aligned} - \Delta _g u = K_g \quad \text{ on } \,\mathbb{R }^2. \end{aligned}$$

(5.4)

Let $\ \varphi : \mathbb{R }^2 \rightarrow G_{n,2} \subseteq \mathbb P ^{n-1}(\mathbb{C })\$ be the Gauß map of $\ f \text{ on } \, \mathbb{R }^2\$ into the Grassmanian of oriented two planes as subset of the complex projective space, see [12, §2.2]. We know from [12, §2.3]

$$\begin{aligned} |D \varphi |^2 = \frac{1}{2} |A|^2, \end{aligned}$$

(5.5)

in particular $D \varphi \in L^2(\mathbb{R }^2)\$ by (5.2), hence $\ \varphi \in W^{1,2}_0(\mathbb{R }^2,\mathbb P ^{n-1}(\mathbb{C }))$ in the sense of [12, §2.1]. Further from [12, §2.2], we know for the standard Kähler two-form $\ \omega \text{ on } \, \mathbb P ^{n-1}(\mathbb{C })\$ that

$$\begin{aligned} \varphi ^* \omega = K_g vol_g. \end{aligned}$$

(5.6)

We get by (5.1)

$$\begin{aligned} \int \limits _{\mathbb{R }^2} \varphi ^* \omega = 0 \end{aligned}$$

and by (5.2)

$$\begin{aligned} \int \limits _{\mathbb{R }^2} J \varphi { \ \mathrm{{d}} }\mu _g \le \frac{1}{2} \int \limits _{\mathbb{R }^2} |D \varphi |^2 { \ \mathrm{{d}} }\mu _g = \frac{1}{4} \int \limits _{\mathbb{R }^2} |A|^2 { \ \mathrm{{d}} }\mu _g \le 2 \pi - \delta /4. \end{aligned}$$

Then by [12, Corollary 3.5.7] there exists a smooth $\ v: \mathbb{R }^2 \rightarrow \mathbb{R }\$ with

$$\begin{aligned} -\Delta v = * \varphi ^* \omega \quad \text{ in } \, \mathbb{R }^2, \lim _{z \rightarrow \infty } v(z) = 0, \end{aligned}$$

(5.7)

and satisfying the estimates

$$\begin{aligned} \Vert v \Vert _{L^\infty (\mathbb{R }^2)}, \Vert D v\Vert _{L^2(\mathbb{R }^2)}, \Vert D^2 v\Vert _{L^1(\mathbb{R }^2)} \!\le \! C(n,\delta ) \int \limits _{\mathbb{R }^2} |D \varphi |^2 { \ \mathrm{{d}} }\mu _g \!\le \! C(n,\delta ) \int \limits _{\mathbb{R }^2} |A|^2 { \ \mathrm{{d}} }\mu _g.\nonumber \\ \end{aligned}$$

(5.8)

We rewrite (5.7) by (5.6) into

$$\begin{aligned} - \Delta _g v = - e^{-2u} \Delta v = e^{-2u} * \varphi ^* \omega = K_g \end{aligned}$$

and see from (5.4) that $\ u - v\$ is an entire harmonic function. But [12, Theorem 4.2.1, Corollary 4.2.5] combined with (5.1), imply that $u$ is bounded. Therefore $\ u - v\$ is also bounded and reduces to a constant $\ \lambda$. Then (5.3) follows from (5.8), which proves the theorem. $\square$

With this theorem, we obtain extensions of [9, Theorems 3.1 and 4.1] along the proofs given there.

Theorem 5.2

Let $\ f: {\Sigma }\rightarrow \mathbb{R }^n\$ be an immersion of a closed, orientable surface $\ {\Sigma } \text{ of } \text{ genus } \,p \ge 1\$ with $\ { \mathcal{W } }(f) \le \Lambda$. Assume that $\ f({\Sigma }) \subseteq \bigcup _{k=1}^K B_{\varrho _k/2}(x_k) \text{ with } \, \varrho _l/\varrho _k \le R$, such that for all $\ k = 1,\ldots , K\$ and some $\ \delta > 0\$ the following conditions hold:

$$\begin{aligned} \int \limits _{B_{\varrho _k}(x_k)} |A|^2 { \ \mathrm{{d}} }\mu < 4 { e_{n} } - \delta = \left\{ \begin{array}{l@{\quad }l@{\quad }l} 16 \pi - \delta &{} \text{ for } &{}n = 3, \\ 32 \pi / 3 - \delta &{} \text{ for } &{}n = 4, \\ 8 \pi - \delta &{} \text{ for }&{}n \ge 5, \\ \end{array} \right. \end{aligned}$$

(5.9)

$$\begin{aligned} \int \limits _{B_{\varrho _k}(x_k) - B_{\varrho _k/2}(x_k)} |A|^2 { \ \mathrm{{d}} }\mu < \varepsilon ^2. \end{aligned}$$

(5.10)

Denoting by $\ D^{k,\alpha }_\sigma , 1 \le \alpha \le m_k$, the components of $\ f^{-1}(B_{\sigma }(x_k))\$ which meet $\ \partial B_{9\varrho _k/16}(x_k)$, we further assume for all $\ \sigma \in [5\varrho _k/8,7\varrho _k/8]\$ up to a set of measure at most $\ \varrho _k/16\$ that

$$\begin{aligned} \int \limits _{D^{k,\alpha }_\sigma } K_g { \ \mathrm{{d}} }\mu _g > -2 \pi + \delta \quad \text{ for } \text{ all }\quad \alpha = 1, \ldots , m_k. \end{aligned}$$

(5.11)

Then for $\ \varepsilon \le \varepsilon (n,\Lambda ,\delta ) \text{ and } \,C_0 \ge C_0(\Lambda )$, there is a constant curvature metric $\ g_0 = e^{-2u} g\$ such that

$$\begin{aligned} \sup \limits _{\Sigma }|u| \le C(n,\Lambda ,R,K,p,\delta ). \end{aligned}$$

$\square$

Remark We see that (5.9) satisfies for $\ n = 3,4\$ the weaker conditions [9] Theorem 3.1 (3.1) and (3.2) with different $\ \delta \$ and for $\ \varepsilon \$ small enough. This follows from $\ |K| \le |A|^2 / 2 \text{ and } |A^0|^2 = |A|^2 / 2 - K\$ and the observation by (5.11) and [9, 2.6] that

$$\begin{aligned} \left| \,\, \int \limits _{D^{k,\alpha }_\sigma } K_g { \ \mathrm{{d}} }\mu _g \right| \le C \varepsilon ^\alpha . \end{aligned}$$

$\square$

We recall the definitions of the constants in (1.1), see also [9],

$$\begin{aligned} \tilde{\beta }^n_p := \min \left\{ 4 \pi + \sum _{i=1}^k (\beta ^n_{p_i}-4\pi ): 1 \le p_i < p,\,\sum _{i=1}^k p_i =p\right\} , \end{aligned}$$

where $\ \tilde{\beta }^n_1 = \infty$, and in (1.3)

$$\begin{aligned} { \mathcal{W } }_{n,p} := \min ( 8 \pi , \tilde{\beta }^n_p , \beta ^n_p + { e_{n} } ). \end{aligned}$$

For $\ n = 3$, the last term could be ommitted as $\ \beta ^3_p + { e_{3} } > 8 \pi$.

Theorem 5.3

For $\ p \ge 1$, let $\ { \mathcal{C } }(n,p,\delta )\$ be the class of closed, orientable, genus $\ p\$ surfaces $\ f: {\Sigma }\rightarrow \mathbb{R }^n\$ satisfying $\ { \mathcal{W } }(f) \le { \mathcal{W } }_{n,p} - \delta \text{ for } \text{ some } \,\delta > 0$. Then for any $\ f \in { \mathcal{C } }(n,p,\delta )\$ there is a Möbius transformation $\ \Phi \$ and a constant curvature metric $\ g_0$, such that the metric $\ g\$ induced by $\ \Phi \circ f\$ satisfies

$$\begin{aligned} g = e^{2u} g_0 \quad \text{ where }\quad \sup \limits _{\Sigma }|u| \le C(n,p,\delta ) < \infty . \end{aligned}$$

$\square$

As already pointed out in [9], the bound $\ { \mathcal{W } }_{3,p} \text{ is } \text{ sharp } \text{ for } \,n = 3$, but for $\ n \ge 4\$ there is no indication that the terms $\ \beta ^4_p + (8 \pi / 3) \text{ or } \,\beta ^n_p + 2 \pi \$ are necessary.

There is also a version to solve (5.7) when $\ f\$ is an immersion of a disc, but we do not use it in the text.

Proposition 5.1

Let $\ f: B_1(0) \subseteq \mathbb{R }^2 \rightarrow \mathbb{R }^n$, be a conformal immersion with induced metric $\ g = e^{2u} { g_{euc} }\$ and square integrable second fundamental form satisfying

$$\begin{aligned} \int \limits _{B_1(0)} |A|^2 { \ \mathrm{{d}} }\mu _g \le \left\{ \begin{array}{l@{\quad }l} 8 \pi - \delta &{} \text{ for }\quad n = 3, \\ 4 \pi - \delta &{} \text{ for }\quad n \ge 4, \\ \end{array} \right. \end{aligned}$$

(5.12)

for some $\ \delta > 0$. Then there exists a smooth solution $\ v: B_1(0) \rightarrow \mathbb{R }\$ of

$$\begin{aligned} - \Delta _g v = K_g \quad \text{ on } \,B_1(0) \end{aligned}$$

(5.13)

satisfying

$$\begin{aligned} \Vert v \Vert _{L^\infty (B_1(0))}, \Vert D v \Vert _{L^2(B_1(0))}, \Vert D^2 v \Vert _{L^1(B_1(0))} \le C(n,\delta ) \int \limits _{B_1(0)} |A|^2 { \ \mathrm{{d}} }\mu _g. \end{aligned}$$

(5.14)

Proof

For $\ n \ge 4$, we let as in the proof of the previous theorem $\ \varphi : B_1(0) \rightarrow G_{n,2} \subseteq \mathbb P ^{n-1}(\mathbb{C })\$ be the Gauß map of $\ f \text{ on } \,B_1(0)\$ into the Grassmanian. Extending by $\ \varphi (z) := \varphi (1/\bar{z}) \text{ for } \,|z| > 1$, we see $\varphi \in W^{1,2}_0(\mathbb{R }^2,\mathbb P ^{n-1}(\mathbb{C }))\$ in the sense of [12, §2.1] by (5.5) and (5.12), and for the standard Kähler two-form $\ \omega \text{ on } \,\mathbb P ^{n-1}(\mathbb{C })$, see [12, §2.2], that $\ \int _{\mathbb{R }^2} \varphi ^* \omega = 0$. Next by (5.5) and (5.12)

$$\begin{aligned} \int \limits _{\mathbb{R }^2} J \varphi { \ \mathrm{{d}} }\mu _g = 2 \int \limits _{B_1(0)} J \varphi { \ \mathrm{{d}} }\mu _g \le \int \limits _{B_1(0)} |D \varphi |^2 { \ \mathrm{{d}} }\mu _g = \frac{1}{2} \int \limits _{B_1(0)} |A|^2 { \ \mathrm{{d}} }\mu _g \le 2 \pi - \delta /2. \end{aligned}$$

Then by [12, Corollary 3.5.7] there exists a smooth $\ v: \mathbb{R }^2 \rightarrow \mathbb{R }\$ with

$$\begin{aligned} -\Delta v = * \varphi ^* \omega \quad \text{ in } \,\mathbb{R }^2, \quad \lim _{z \rightarrow \infty } v(z) = 0, \end{aligned}$$

(5.15)

and satisfying the estimates

$$\begin{aligned} \Vert v \Vert _{L^\infty (\mathbb{R }^2)}, \Vert D v\Vert _{L^2(\mathbb{R }^2)}, \Vert D^2 v\Vert _{L^1(\mathbb{R }^2)} \le C(n,\delta ) \int \limits _{\mathbb{R }^2} |D \varphi |^2 { \ \mathrm{{d}} }\mu _g \le C(n,\delta ) \int \limits _{B_1(0)} |A|^2 { \ \mathrm{{d}} }\mu _g.\nonumber \\ \end{aligned}$$

(5.16)

As above (5.15) implies by (5.6) that $\ - \Delta _g v = K_g \text{ on } \, B_1(0)$, hence (5.13), and (5.16) gives (5.14), which proves the propsition for $\ n \ge 4$.

We improve for $\ n = 3\$ by considering the Gauß map of $\ f \text{ on } \,B_1(0)\$ into the sphere as $\ \nu : B_1(0) \rightarrow S^2$. Extending as above by $\ \nu (z) := \nu (1/\bar{z}) \text{ for } \,|z| > 1$, we see $\nu \in W^{1,2}_0(\mathbb{R }^2,S^2) \text{ and } \,\int _{\mathbb{R }^2} \nu ^* vol_{S^2} = 0$. Next $\ J \nu = |K_g|\$ and by (5.12)

$$\begin{aligned} \int \limits _{\mathbb{R }^2} J \nu { \ \mathrm{{d}} }\mu _g = 2 \int \limits _{B_1(0)} |K_g| { \ \mathrm{{d}} }\mu _g \le \int \limits _{B_1(0)} |A|^2 { \ \mathrm{{d}} }\mu _g < 8 \pi - \delta . \end{aligned}$$

Then proceeding as in [9, Theorem 6.1] there exists a smooth $\ v: \mathbb{R }^2 \rightarrow \mathbb{R }\$ with

$$\begin{aligned} -\Delta v = * \nu ^* vol_{S^2} \quad \text{ in } \,\mathbb{R }^2, \quad \lim _{z \rightarrow \infty } v(z) = 0, \end{aligned}$$

and satisfying the estimates

$$\begin{aligned} \Vert v \Vert _{L^\infty (\mathbb{R }^2)}, \Vert D v\Vert _{L^2(\mathbb{R }^2)}, \Vert D^2 v\Vert _{L^1(\mathbb{R }^2)} \le C(\delta ) \int \limits _{\mathbb{R }^2} |D \nu |^2 { \ \mathrm{{d}} }\mu _g \le C(\delta ) \int \limits _{B_1(0)} |A|^2 { \ \mathrm{{d}} }\mu _g. \end{aligned}$$

Observing that $\ * \nu ^* vol_{S^2} = K_g \text{ on } \,B_1(0)$, this proves the proposition for $\ n = 3\$ as above. $\square$

Remark We mention that (5.12) can be replaced along [9] Theorem 6.1 by

$$\begin{aligned}&\displaystyle \int \limits _{B_1(0)} |K_g| { \ \mathrm{{d}} }\mu _g \le 4 \pi - \delta \quad \text{ for } \quad n = 3,\nonumber \\&\left. \begin{array}{l} {\int \nolimits _{B_1(0)} |K_g| { \ \mathrm{{d}} }\mu _g + \frac{1}{2} \int _{B_1(0)} |A^0|^2 { \ \mathrm{{d}} }\mu _g} \le 4 \pi - \delta , \\ {\int \nolimits _{B_1(0)} |A^0|^2 { \ \mathrm{{d}} }\mu _g} < 4 \pi , \end{array} \right\} \quad \text{ for } \quad n = 4. \end{aligned}$$

$\square$

Appendix B: Convergence in $W^{2,2}$

In this appendix, we prove a useful convergence proposition.

Proposition 6.1

Let $\ f_m: {\Sigma }\rightarrow \mathbb{R }^n\$ be smooth immersions of a closed, orientable surface $\ {\Sigma }\$ uniformly conformal to some smooth unit volume constant curvature metrics $\ { g_{{ poin },m} }\$ and

$$\begin{aligned}&f_m^* { g_{euc} }= e^{2 u_m} { g_{{ poin },m} }, \nonumber \\&{ \mathcal{W } }(f_m), \parallel u_m \parallel _{L^\infty ({\Sigma })} \le \Lambda \end{aligned}$$

for some $\ \Lambda < \infty$.

Then for a subsequence there exist diffeomorphisms $\ \phi _m: {\Sigma }\stackrel{\approx }{\longrightarrow } {\Sigma }\$ such that replacing $\ f_m \text{ by } \,f_m \circ \phi _m\$

$$\begin{aligned}&f_m \rightarrow f \text{ weakly } \text{ in } \,W^{2,2}({\Sigma }), \text{ weakly }^* \text{ in } \,W^{1,\infty }({\Sigma }), \nonumber \\&u_m \rightarrow u \text{ weakly } \text{ in } \,W^{1,2}({\Sigma }), \text{ weakly }^* \text{ in } \,L^\infty ({\Sigma }), \\&{ g_{{ poin },m} }\rightarrow { g_{{ poin }} }\quad \text{ smoothly }, \nonumber \end{aligned}$$

(6.1)

$$\begin{aligned} f^* { g_{euc} }= e^{2u} { g_{{ poin }} }. \end{aligned}$$

(6.2)

If

$$\begin{aligned} { \pi }(f_m^* { g_{euc} }) \rightarrow \tau \text{ in } \,{ \mathcal T }, \end{aligned}$$

(6.3)

then $\ \phi _m\$ can be chosen with $\ \phi _m \simeq id_{\Sigma }\$ and

$$\begin{aligned} { \pi }(f^* { g_{euc} }) = { \pi }({ g_{{ poin }} }) = \tau . \end{aligned}$$

Proof

By [9, Lemma 5.1] the conformal structures induced by $\ g_m := f_m^* { g_{euc} } \text{ respectively } \text{ by } \,{ g_{{ poin },m} }\$ are compactly contained in moduli space, hence there exist diffeomorphisms $\ \phi _m: {\Sigma }\stackrel{\approx }{\longrightarrow } {\Sigma }\$ such that for a subsequence

$$\begin{aligned} \phi _m^* { g_{{ poin },m} }\rightarrow { g_{{ poin }} }\quad \text{ smoothly } \end{aligned}$$

to a smooth unit volume constant curvature metric $\ { g_{{ poin }} }$. After reparametrizing, we may assume $\ \phi _m = id_{\Sigma }$. We get by elementary differential geometry and the Gauß–Bonnet theorem

$$\begin{aligned} -\Delta _{g_m} u_m + 2 \pi \chi ({\Sigma }) e^{-2u_m} = K_{g_m}. \end{aligned}$$

(6.4)

Therefore

$$\begin{aligned}&\int \limits _{\Sigma }|D u_m|^2_{ g_{{ poin },m} }{ \ \mathrm{{d}} }\mu _{ g_{{ poin },m} }= \int \limits _{\Sigma }|D u_m|^2_{g_m} { \ \mathrm{{d}} }\mu _{g_m}\\&\quad = -\int \limits _{\Sigma }\Delta _{g_m} u_m \cdot u_m{ \ \mathrm{{d}} }\mu _{g_m} =\int \limits _{\Sigma }\left( K_{g_m} - 2 \pi \chi ({\Sigma }) e^{-2u_m} \right) u_m { \ \mathrm{{d}} }\mu _{g_m} \nonumber \\&\quad \le C({\Sigma },\Lambda ) \left( \int \limits _{\Sigma }1 { \ \mathrm{{d}} }\mu _{ g_{{ poin },m} }+ \int \limits _{\Sigma }|K_{g_m}| { \ \mathrm{{d}} }\mu _{g_m} \right) \le C({\Sigma },\Lambda ) \left( 1 + \int \limits _{\Sigma }|A_{f_m}|^2 { \ \mathrm{{d}} }\mu _{f_m} \right) \nonumber \\&\quad \le C({\Sigma },\Lambda ) (1 + { \mathcal{W } }(f_m)) \le C({\Sigma },\Lambda ), \end{aligned}$$

as $\ { \mathcal{W } }(f_m), |u_m| \le \Lambda , |K| \le |A|^2/2\$ and the Gauß–Bonnet theorem. Therefore

$$\begin{aligned} \parallel D u_m \parallel _{L^2({\Sigma },{ g_{{ poin },m} })} \le C({\Sigma },\Lambda ), \end{aligned}$$

and, as $\ { g_{{ poin },m} }\rightarrow { g_{{ poin }} }$, we get for a subsequence

$$\begin{aligned} u_m \rightarrow u \left\{ \begin{array}{l} \text{ weakly } \text{ in } \,W^{1,2}({\Sigma }), \\ \text{ weakly }^* \text{ in } \,L^\infty ({\Sigma }), \\ \text{ and } \text{ pointwise } \text{ almost } \text{ everywhere } \text{ on } \,{\Sigma }. \\ \end{array} \right. \end{aligned}$$

Next

$$\begin{aligned} \Delta _{ g_{{ poin },m} }f_m = e^{2 u_m} \overrightarrow{\mathbf{H }}_{f_m} \quad \text{ on } \,{\Sigma }\end{aligned}$$

and, as $\ |u_m| \le \Lambda$,

$$\begin{aligned} \int \limits _{\Sigma }|e^{2 u_m} \overrightarrow{\mathbf{H }}_{f_m}|^2 { \ \mathrm{{d}} }\mu _{ g_{{ poin },m} }\le C(\Lambda ) \int \limits _{\Sigma }|\overrightarrow{\mathbf{H }}_{f_m}|^2 { \ \mathrm{{d}} }\mu _{f_m} = C(\Lambda ) { \mathcal{W } }(f_m) \le C(\Lambda ), \end{aligned}$$

and get by standard elliptic theory, see [5] Theorem 8.8, and $\ { g_{{ poin },m} }\rightarrow { g_{{ poin }} }\$ smoothly that $\ f_m \text{ is } \text{ bounded } \text{ in } \,W^{2,2}({\Sigma })$, hence after passing to a subsequence

$$\begin{aligned} f_m \rightarrow f \text{ weakly } \text{ in } \,W^{2,2}({\Sigma }). \end{aligned}$$

As $\ f_m^* { g_{euc} }= e^{2 u_m} { g_{{ poin },m} } \text{ and } \,|u_m| \le \Lambda , { g_{{ poin },m} }\rightarrow { g_{{ poin }} }$, we see that $\ \nabla f_m \text{ is } \text{ bounded } \text{ in } \, L^\infty ({\Sigma })$, hence $\ \nabla f_m \rightarrow \nabla f \text{ is } \text{ weakly }^* \text{ in } \,L^\infty ({\Sigma })$. By the above convergences

$$\begin{aligned} f^* { g_{euc} }= e^{2u} { g_{{ poin }} }. \end{aligned}$$

If $\ { \pi }(f_m^* { g_{euc} }) \rightarrow \tau \$ in Teichmüller space, we may further assume that $\ \phi _m \simeq id_{\Sigma }\$ and

$$\begin{aligned} \tau \leftarrow { \pi }(f_m^* { g_{euc} }) = { \pi }({ g_{{ poin },m} }) \rightarrow { \pi }({ g_{{ poin }} }), \end{aligned}$$

hence $\ { \pi }(f^* { g_{euc} }) = { \pi }({ g_{{ poin }} }) = \tau$, and the proposition is proved. $\square$

Remarks   1. Clearly (6.1) is equivalent to [10, (2.3) – (2.5)], (6.2) is [10, 2.6], and (6.3) is [10, 2.2] with $\ \tau _0 \text{ replaced } \text{ by } \, \tau$.

• 2. Convergence of the Willmore energy

  $$\begin{aligned} { \mathcal{W } }(f_m) \rightarrow { \mathcal{W } }(f) \end{aligned}$$

  gives even strong convergence $\ f_m \rightarrow f \text{ in } \, W^{2,2}$, see [10, Proposition 5.3].

• 3. If we add the assumption that $\ { g_{{ poin },m} }\rightarrow { g_{{ poin }} }\$ smoothly, the statement of the proposition is true without diffeomorphisms $\ \phi _m$, as it is immediately seen from the beginning of the proof.

$\square$

Appendix C: $W^{2,2}$-immersions

Proposition 7.1

Let $\ f: B_1(0) \subseteq \mathbb{R }^2 \rightarrow \mathbb{R }^n\$ with uniformly positive definite pull-back metric in the sense that

$$\begin{aligned} c_0 { g_{euc} }\le g := f^* { g_{euc} }\le C { g_{euc} }\end{aligned}$$

(7.1)

for some $\ 0 < c_0 \le C < \infty$.

Then for $\ \mu _f := f(\mu _g)\$

$$\begin{aligned} \# f^{-1}(x) \le (2 C / c_0)\ \theta ^2_*(\mu _f,x) \quad \text{ for } \text{ any } \;x \in \mathbb{R }^n, \end{aligned}$$

(7.2)

and if further $\ f \in W^{2,2}(B_1(0))$, then

$$\begin{aligned} \# f^{-1}(x) \le \theta ^2(\mu _f,x) \quad \text{ for } \text{ all } \ x \in \mathbb{R }^n. \end{aligned}$$

(7.3)

Proof

We consider finitely many distinct $\ p_1, \ldots , p_N \in f^{-1}(x)\$ and see for $\ \varrho \$ small that $\ B_\varrho (p_k) \subseteq B_1(0)\$ are pairwise disjoint. By (7.1), we get $\ lip\ f \le \sqrt{2 C} < \infty \$ and

$$\begin{aligned} \mu _f(B_{L \varrho }(x)) = \mu _g \left( f^{-1}(B_{L \varrho }(x)) \right) \ge \mu _g( \cup _{i = 1}^N B_\varrho (p_i) ) \ge c_0 N \pi \varrho ^2, \end{aligned}$$

hence

$$\begin{aligned} \theta ^2_*(\mu _f,x) = \liminf \limits _{\varrho \rightarrow 0} \mu _f(B_{\varrho }(x)) / (\pi \varrho ^2) \ge c_0 N / L^2, \end{aligned}$$

which is (7.2).

To proceed for $\ f \in W^{2,2}$, it suffices to consider $\ \theta ^2_*(\mu _f,x) < \infty$, in particular $\ f^{-1}(x)\$ is finite. We see for $\ \varrho \$ small enough that $\ x \not \in f(\partial B_\varrho (p_i))\$ and claim for $\ \mu _i := f( \mu _g \lfloor B_\varrho (p_i))\$ that

$$\begin{aligned} \theta ^2(\mu _i,x) \ge 1. \end{aligned}$$

(7.4)

This implies

$$\begin{aligned} N \le \sum \limits _{i=1}^N \theta ^2(\mu _i,x) \le \theta ^2(\mu _f,x), \end{aligned}$$

and (7.3) follows.

To prove (7.4), we consider $\ f(0) = 0 \text{ with } \,\liminf _{|p| \rightarrow 1} |f(p)| \ge r > 0$. As $\ f\$ is lipschitz, the image $\ \mu _f = f(\mu _g) = f(J_{g_0} f \cdot \mu _{g_0})\$ is an integral varifold, see [15, 15.7]. We prove that $\ \mu _f\$ has square integrable weak mean curvature in $\ B_r(0)$. For $\ \eta \in C^1_0(B_r(0),\mathbb{R }^n)$, we see that $\ \eta \circ f\$ has compact support in $\ B_1(0)\$ and calculate the first variation

$$\begin{aligned} \delta \mu _f(\eta ) = \int div_{\mu _f} \eta { \ \mathrm{{d}} }\mu _f = \int \limits _{B_1(0)} (div_{\mu _f} \eta ) \circ f { \ \mathrm{{d}} }\mu _g, \end{aligned}$$

where the divergence is given by

$$\begin{aligned} (div_{\mu _f} \eta ) \circ f = g^{ij} \partial _i f^T ((D \eta ) \circ f) \partial _j f = g^{ij} \langle \partial _i f , \partial _j (\eta \circ f) \rangle . \end{aligned}$$

We continue

$$\begin{aligned} \delta \mu _f(\eta )&= \int \limits _{B_1(0)} g^{ij} \langle \partial _i f , \partial _j (\eta \circ f) \rangle \sqrt{g} { \ \mathrm{{d}} }{ \mathcal{L }^2 }= - \int \limits _{B_1(0)} \partial _j( g^{ij} \sqrt{g} \partial _i f) (\eta \circ f) { \ \mathrm{{d}} }{ \mathcal{L }^2 }\nonumber \\&= - \int \limits _{B_1(0)} (\Delta _g f) (\eta \circ f) { \ \mathrm{{d}} }\mu _g = - \int \limits _{B_1(0)} \overrightarrow{\mathbf{H }}_f (\eta \circ f) { \ \mathrm{{d}} }\mu _g. \end{aligned}$$

Therefore $\ \mu _f\$ has weak mean curvature in $\ B_r(0)\$ given by

$$\begin{aligned} \overrightarrow{\mathbf{H }}_{\mu _f} = \sum \limits _{p \in f^{-1}} \overrightarrow{\mathbf{H }}_f(p) \in L^2(\mu _f). \end{aligned}$$

(7.5)

Then from [8, A.10], we get $\ \theta ^2(\mu _f,x) \ge 1\$ for any $\ x \in spt\ \mu _f \cap B_r(0)\$. As $\ f(0) = 0$, we get from (7.2) that $\ \theta ^2_*(\mu _f,0) > 0$, in particular $\ 0 \in { spt\ }\ \mu _f$, and (7.4) follows. $\square$

Proposition 7.2

Let $\ f \in W^{2,2}({\Sigma },\mathbb{R }^n)\$ for some closed surface $\ {\Sigma }\$ with uniformly positive definite pull-back metric in the sense that

$$\begin{aligned} c_0 g_0 \le g := f^* { g_{euc} }\le C g_0 \end{aligned}$$

(7.6)

for some smooth metric $\ g_0 \text{ on } \,{\Sigma } \text{ and } \,0 < c_0 \le C < \infty$.

Then

$$\begin{aligned} \mu _f&:= f(\mu _g) = \# f^{-1} \cdot { \mathcal{H }^2 }\lfloor f({\Sigma }) = \# f^{-1} \cdot { \mathcal{H }^2 }\lfloor spt\ \mu _f, \nonumber \\ spt\ \mu _f&= f({\Sigma }). \end{aligned}$$

(7.7)

Moreover $\ \mu _f\$ is an integral varifold with square integrable weak mean curvature and

$$\begin{aligned} { \mathcal{W } }(\mu _f)&= { \mathcal{W } }(f), \nonumber \\ \# f^{-1}(x)&\le \theta ^2(\mu _f,x) \le { \mathcal{W } }(f) / (4 \pi ) \quad \text{ for } \text{ all } \,\;\;\;x \in \mathbb{R }^n. \end{aligned}$$

(7.8)

If $\ { \mathcal{W } }(f) < 8 \pi$, then $\ f\$ is injective, in particular

$$\begin{aligned} f: {\Sigma }\,{ \stackrel{\approx }{\longrightarrow } }\, spt\ \mu _f = f({\Sigma }) \end{aligned}$$

is a homeomorphism and

$$\begin{aligned} \mu _f = { \mathcal{H }^2 }\lfloor spt\ \mu _f = { \mathcal{H }^2 }\lfloor f({\Sigma }). \end{aligned}$$

If moreover $\ f\$ is uniformly conformal to a smooth metric $\ g_0 \text{ on } \,{\Sigma }$, that is $\ f^* { g_{euc} }= e^{2u} g_0 \text{ with } \, u \in L^\infty ({\Sigma })$, then $\ f\$ is a bi-lipschitz homeomorphism.

Proof

Again as $\ f\$ is lipschitz from (7.6), we see that the image $\ \mu _f = f(\mu _g) = f(J_{g_0} f \cdot \mu _{g_0})\$ is an integral varifold and

$$\begin{aligned} \mu _f = \# f^{-1} \cdot { \mathcal{H }^2 }\lfloor f({\Sigma }), \end{aligned}$$

(7.9)

see [15, 15.7]. Clearly $\ spt\ \mu _f \subseteq f({\Sigma })$, as $\ f({\Sigma })\$ is compact, hence closed. On the other hand by (7.2), we get $\ \theta ^2_*(\mu _f) > 0 \text{ on } \, f({\Sigma })$, hence $\ spt\ \mu _f = f({\Sigma })$, and (7.7) follows.

As $\eta \circ f$ has compact support in ${\Sigma }$ for any $\ \eta \in C^1_0(\mathbb{R }^{n},\mathbb{R }^{n})$, we see as in (7.5) that $\ \mu _f\$ has weak mean curvature given by

$$\begin{aligned} \overrightarrow{\mathbf{H }}_{\mu _f} = \sum \limits _{p \in f^{-1}} \overrightarrow{\mathbf{H }}_f(p) \in L^2(\mu _f) \end{aligned}$$

and

$$\begin{aligned} { \mathcal{W } }(\mu _f) = \frac{1}{4} \int |\overrightarrow{\mathbf{H }}_{\mu _f}|^2 { \ \mathrm{{d}} }\mu _f = \frac{1}{4} \int \limits _{\Sigma }|\overrightarrow{\mathbf{H }}_f|^2 { \ \mathrm{{d}} }\mu _g = { \mathcal{W } }(f) < \infty , \end{aligned}$$

which gives the first part in (7.8). Then from [8, A.17], we get $\ \theta ^2(\mu _f) \le { \mathcal{W } }(\mu _f) / (4 \pi )$, and the second part follows from (7.3).

If $\ { \mathcal{W } }(f) < 8 \pi$, we see that $\ \# f^{-1} \le 1$, hence $\ f\$ is injective and by (7.7)

$$\begin{aligned} \mu _f = { \mathcal{H }^2 }\lfloor spt\ \mu _f = { \mathcal{H }^2 }\lfloor f({\Sigma }). \end{aligned}$$

As obviusly $\ f: {\Sigma }\rightarrow spt\ \mu _f = f({\Sigma })\$ is surjective, we get that $\ f\$ is bijective, and, as $\ f\$ is continuous and $\ {\Sigma }\$ is compact, we see that $\ f\$ is a homeomorphism.

Finally we assume $\ f\$ to be uniformly conformal. We already know that $\ f\$ is lipschitz, and it remains to prove that its inverse is lipschitz. We will use [12, Lemma 4.2.8]. If $\ f^{-1}\$ is not lipschitz, we have $\ p_k,q_k \in {\Sigma }\$ with

$$\begin{aligned} |f(p_k) - f(q_k)| < d_{{\Sigma },g_0}(p_k,q_k) / k. \end{aligned}$$

(7.10)

We get for a subsequence $\ p_k \rightarrow p, q_k \rightarrow q \text{ in } \,{\Sigma }$, and, as $\ diam_{g_0} {\Sigma }\$ is finite, $\ f(p) = f(q)$, hence $\ p = q$, as $\ f\$ is injective. Introducing local conformal coordinates for $\ g_0\$ around $p$, we get an open neighbourhood $\ U(p) \text{ of } \,p\$ and $\ \varphi : B_1(0) \cong U(p) \text{ with } \,\varphi (0) = p, \varphi ^* g_0 = e^{2v} { g_{euc} }, v \in L^\infty (B_1(0))$. Then

$$\begin{aligned} (f \circ \varphi )^* { g_{euc} }= \varphi ^*(e^{2u} g_0) = e^{2 (v + u \circ \varphi )} { g_{euc} }. \end{aligned}$$

and $\ f_0 := f \circ \varphi : B_1(0) \rightarrow \mathbb{R }^n\$ is conformal and $\ f_0 \in W^{2,2}_{loc}(B_1(0))$. Then for $\ M := \parallel v \parallel _{L^\infty (B_1(0))} + \parallel u \parallel _{L^\infty ({\Sigma })} < \infty$, we can choose $\ 0 < \varrho < 1\$ with

$$\begin{aligned} \int \limits _{B_\varrho (0)} |D^2 f_0|^2 { \ \mathrm{{d}} }{ \mathcal{L }^2 }< c_0 e^{-2 M}, \end{aligned}$$

where $\ c_0 = (\pi \tanh \pi ) / 2\$ is given in [12, Lemma 4.2.8]. For $\ k\$ large, a square with vertices $\ \varphi ^{-1}(p_k), \varphi ^{-1}(q_k)\$ is contained in $\ B_\varrho (0)$, and we get from [12, Lemma 4.2.8]

$$\begin{aligned} d_{{\Sigma },g_0}(p_k,q_k) \le e^M d_{{\Sigma },f^* { g_{euc} }}(p_k,q_k) \le \sqrt{2} e^M |f(p_k) - f(q_k)|, \end{aligned}$$

which contradicts (7.10), and the proposition is proved. $\square$