On the Negativity of Ricci Curvatures of Complete Conformal Metrics

Abstract

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases.

1 Introduction

Let (M, g) be a compact Riemannian manifold of dimension n without boundary, for $n\ge 3$, and $\Gamma$ be a smooth submanifold in M. For $(M,g)=(S^n, g_{S^n})$, Loewner and Nirenberg [15] proved that there exists a complete conformal metric on $S^n \setminus \Gamma$ with a negative constant scalar curvature if and only if dim$(\Gamma )>(n-2)/2$. Aviles and McOwen [4] proved a similar result for the general manifold (M, g). As a consequence, we can take the dimension of the submanifold to be $n-1$ and conclude the following result: In any compact Riemannian manifold with boundary, there exists a complete conformal metric with a negative constant scalar curvature. See [4]. For convenience, we always take the constant scalar curvature to be $-n(n-1)$. In this paper, we will study whether Ricci curvatures of such a metric remain negative.

For the case of positive scalar curvatures, the existence and asymptotic behaviors of solutions have been extensively studied over the years. We shall not discuss this case here, but refer to [5, 12, 19,20,21, 23].

There are several classical results for metrics with negative Ricci curvatures. Gao and Yau [7] proved that there exists a metric of negative Ricci curvature on every compact 3-dimensional manifold without boundary. Lohkamp [16] generalized this to arbitrary dimensions and proved that any manifold of dimension $n\ge 3$ (compact or not) admits a complete metric of negative Ricci curvature. Restricted to conformal metrics, by solving $\det (\mathrm{Ric})= \mathrm{constant}$ with a precise boundary asymptotics, Guan [8] and Gursky, Streets and Warren [9] proved that there exists a complete conformal metric with negative Ricci curvature on a compact Riemannian manifold with boundary.

In the unit ball in the Euclidean space, the complete conformal metric with scalar curvature $-n(n-1)$ is exactly the Poincaré metric of the unit ball model of the hyperbolic space and has sectional curvatures $-1$ and Ricci curvatures $-(n-1)$. In particular, it has negative sectional curvatures and Ricci curvatures. A natural question is whether this remains true for the more general case; namely, whether the complete conformal metric with a negative constant scalar curvature in a compact Riemannian manifold with boundary has negative sectional curvatures or negative Ricci curvatures. We point out that a straightforward calculation based on the polyhomogeneous expansion established in [1] and [18] yields that such a metric has sectional curvatures asymptotically equal to $-1$ near boundary. Our main concern is whether the negativity of the sectional curvatures or Ricci curvatures near boundary can be carried over to the entire domain.

In view of the Poincaré metric in the unit ball model of the hyperbolic space, it is reasonable to expect that the complete conformal metric with a negative constant scalar curvature should have negative sectional curvatures in a domain close to the unit ball in the Euclidean space. We will confirm this in this paper. In fact, we will prove an affirmative result for convex domains in the Euclidean space.

Theorem 1.1

Let $\Omega \subset {\mathbb {R}}^{n}$ be a bounded convex domain, for $n\ge 3$, and $g_\Omega$ be the complete conformal metric in $\Omega$ with the constant scalar curvature $-n(n-1)$. Then, $g_\Omega$ has negative sectional curvatures in $\Omega$. Moreover, $g_\Omega$ has Ricci curvatures strictly less than $-{n}/{2}$ in $\Omega$.

The convexity assumption of the domain $\Omega$ is crucial. It allows us to apply a convexity theorem by Kennington [10] directly to conformal factors. Theorem 1.1 does not hold for general bounded domains in ${\mathbb {R}}^n$. In fact, in certain bounded star-shaped domains, the conformal metrics may have arbitrarily large positive Ricci curvature components. See Example 5.6. Note that bounded convex domains and bounded star-shaped domains have the same topology.

Closely related to the negativity of the Ricci curvatures is whether there is a constant rank theorem for metrics with negative Ricci curvatures, since it is already known that the Ricci curvatures are negative near boundary. Caffarelli, Guan, and Ma [6] proved a constant rank theorem for the $\sigma _k$-curvature equations under certain positivity conditions on curvatures. However, their result is not applicable in our case. Our strategy is to connect directly boundary curvatures of domains with the interior curvature tensors of the complete conformal metrics.

We now turn our attention to bounded smooth domains which are sufficiently “far” from the unit ball. According to Aviles and McOwen [4], to have a complete conformal metric with constant negative scalar curvature in $M\setminus \Gamma$, it is required that dim$(\Gamma )>(n-2)/2$. Closely related is a result proved by Mazzeo and Pacard [19] that there exist complete conformal metrics in $S^n\setminus \Gamma$ with constant positive scalar curvatures if dim$(\Gamma )\le (n-2)/2$. In view of these results, we can ask what happens to Ricci curvatures of the complete conformal metrics with scalar curvatures fixed at $-n(n-1)$ in domains $\Omega \subset M$ whose $(n-1)$-dimensional boundary is close to a smooth submanifold $\Gamma$ of dimension $\le (n-2)/2$. Do Ricci curvatures have mixed signs as $\partial \Omega$ becomes close to a low dimensional set, say a single point?

In this paper, we will construct domains where complete conformal metrics have large positive Ricci curvatures in domains in compact Riemannian manifolds.

Theorem 1.2

Let (M, g) be a compact Riemannian manifold of dimension $n\ge 3$ without boundary and $\Gamma$ be a disjoint union of finitely many closed smooth embedded submanifolds in M of varying dimensions, between 0 and $(n-2)/2$. Consider the following cases:

Case 1.:

$\Gamma$ contains a submanifold of dimension j, with $1\le j\le (n-2)/2$.

Case 2.:

If (M, g) is not conformally equivalent to the standard sphere $S^n$, $\Gamma$ consists of finitely many points.

Case 3.:

If (M, g) is conformally equivalent to $S^{n}$, $\Gamma$ consists of at least two but only finitely many points.

Suppose that $\Omega _i$ is a sequence of increasing domains with smooth boundary in M which converges to $M \setminus \Gamma$ and that $g_i$ is the complete conformal metric in $\Omega _i$ with the constant scalar curvature $-n(n-1)$. Then, for sufficiently large i, $g_i$ has a positive Ricci curvature component somewhere in $\Omega _{i}$. Moreover, the maximal Ricci curvature in $\Omega _{i}$ diverges to $\infty$ as $i\rightarrow \infty$.

By the convergence of $\Omega _i$ to $M \setminus \Gamma$, we mean $\textstyle\bigcup _{i=1}^\infty \Omega _i =M\setminus \Gamma$ and, for any $\varepsilon >0$, $\partial \Omega _i$ is in the $\varepsilon$-neighborhood of $\Gamma$ for all large i. By convention, a zero dimensional submanifold is simply a point.

The difference between Case 2 and Case 3 in Theorem 1.2 lies on the number of isolated points when closed smooth embedded submanifolds of positive dimension are absent from $\Gamma$. On manifolds conformally equivalent to the standard sphere, the number of the isolated points has to be at least two; while on manifolds not conformally equivalent to the standard sphere, we can allow one point. Theorem 1.2 does not necessarily hold if $\Gamma$ consists of one point on manifolds conformally equivalent to the standard sphere. See Remark 4.5. Such a difference demonstrates that the background manifolds also play a decisive role in the issue studied in this paper.

As a consequence of Case 2, with $\Gamma$ consisting of just one point, we have the following rigidity result.

Theorem 1.3

Let (M, g) be a compact Riemannian manifold of dimension $n\ge 3$ without boundary and $x_0$ be a point in M. Suppose that there exists a sequence $\Omega _i$ of increasing domains with smooth boundary in M which converges to $M {\setminus }\{x_0\}$, such that the complete conformal metric in $\Omega _i$ with the constant scalar curvature $-n(n-1)$ has uniformly bounded Ricci curvatures in $\Omega _{i}$. Then, M is conformally equivalent to the standard sphere $S^n$.

The set $\Gamma$ in Theorem 1.2 resembles that in [19]. The metric $g_i$ in $\Omega _i$ as in Theorem 1.2 is assumed to have a negative constant scalar curvature, $-n(n-1)$. As $\Omega _i$ becomes close to $M\setminus \Gamma$, Ricci curvatures split in sign. Some components become negatively large, while some others positively large.

The proof of Theorem 1.2 relies on a careful analysis of the Ricci curvatures of the complete conformal metrics near boundary. The polyhomogeneous expansion provides correct values near boundary for applications of the maximum principle. The Yamabe invariant of (M, g) plays a crucial role and determines behaviors of the convergence of the conformal factors. Among the three cases listed in Theorem 1.2, Case 2 is the most difficult to prove, especially when the Yamabe invariant is between zero and that of the standard sphere. When $\Gamma$ consists of one point $x_0$, we need expansions of Green’s functions. If $n=3,4,5$, or M is conformally flat in a neighborhood of $x_0$, we need to employ the positive mass theorem. If $n\ge 6$ and M is not conformally flat in a neighborhood of $x_0$, we need to distinguish the two cases $W(x_0)\ne 0$ and $W(x_0)=0$. Discussions for the latter case is much more complicated than the former case. The proof here seems to resemble the solution of the Yamabe problem, but with one twist. In solving the Yamabe problem, we can choose a point where the Weyl tensor is not zero in the case that M is not conformally flat. In our case, $x_0$ is a given point and the Weyl tensor can be zero even if M is not conformally flat in a neighborhood of $x_0$. Different vanishing orders of W at $x_0$ requires different methods. In fact, we also need to employ the positive mass theorem if the Weyl tensor vanishes at $x_0$ up to a sufficiently high order. The positive mass theorem has been known to be true if $3\le n\le 7$, or M is locally conformally flat, or M is spin. See [13, 22, 24] and [26]. These conditions might be technical and could be removed according to the recent papers [17] and [25]. Refer to Remark 4.4 on how the positive mass theorem is used in the proof of Theorem 1.2.

The paper is organized as follows. In Sect. 2, we discuss some preliminary identities. In Sect. 3, we study the Ricci curvatures of complete conformal metrics in bounded convex domains in the Euclidean space and prove Theorem 1.1. In Sect. 4, we study the Ricci curvatures of complete conformal metrics in domains in compact manifolds and prove Theorem 1.2. In Sect. 5, we present several examples in the Euclidean space.

We would like to thank Matthew Gursky for suggesting the problem studied in this paper and many helpful discussions. Gursky graciously shared many of his stimulating computations with us. We would also like to thank Yuguang Shi for helpful discussions.

2 Preliminaries

Let (M, g) be a smooth Riemannian manifold of dimension n, for some $n\ge 3$, either compact without boundary or noncompact and complete. Assume $\Omega \subset M$ is a smooth domain, with an $(n-1)$-dimensional boundary. If (M, g) is noncompact, we assume, in addition, that $\Omega$ is bounded. We consider the following problem:

$$\begin{aligned} \Delta _{g} u -\frac{n-2}{4(n-1)}S_gu&= \frac{1}{4}n(n-2) u^{\frac{n+2}{n-2}} \quad \text {in }\,\Omega , \end{aligned}$$

(2.1)

$$\begin{aligned} u&=\infty \quad \text {on }\partial \Omega , \end{aligned}$$

(2.2)

where $S_g$ is the scalar curvature of M. According to Loewner and Nirenberg [15] for $(M,g)=(S^n, g_{S^n})$ and Aviles and McOwen [4] for the general case, (2.1) and (2.2) admits a unique positive solution. We note that $u^{\frac{4}{n-2}}g$ is the complete metric with a constant scalar curvature $-n(n-1)$ on $\Omega$. Andersson, Chruściel and Friedrich [1] and Mazzeo [18] established the polyhomogeneous expansions for the solutions. For the first several terms, we have

$$\begin{aligned} u=d^{-\frac{n-2}{2}}\bigg [1+\frac{n-2}{4(n-1)}H_{\partial \Omega }d+O(d^2)\bigg ], \end{aligned}$$

where d is the distance to $\partial \Omega$ and $H_{\partial \Omega }$ is the mean curvature of $\partial \Omega$ with respect to the interior unit normal vector of $\partial \Omega$. Set

$$\begin{aligned} v=u^{-\frac{2}{n-2}}.\end{aligned}$$

(2.3)

Then,

$$\begin{aligned} v\Delta _{g} v +\frac{1}{2(n-1)}S_{ g} v^2&= \frac{n}{2}(|\nabla _{g} v|^2-1) \quad \text {in }\Omega , \end{aligned}$$

(2.4)

$$\begin{aligned} v&=0\quad \text {on }\partial \Omega . \end{aligned}$$

(2.5)

Moreover,

$$\begin{aligned} v=d-\frac{1}{2(n-1)}H_{\partial \Omega }d^2+O(d^3). \end{aligned}$$

(2.6)

This implies

$$\begin{aligned} |\nabla _gv|=1\quad \text {on }\partial \Omega . \end{aligned}$$

(2.7)

We will use this repeatedly later on.

Consider the conformal metric

$$\begin{aligned} g_\Omega =u^{\frac{4}{n-2}}g=v^{-2}g.\end{aligned}$$

(2.8)

For a unit vector X of g, vX is a unit vector of $g_\Omega$. Let $R_{ij}$ be the Ricci components of g in a local frame for the metric g and $R^\Omega _{ij}$ be the Ricci components of $g_\Omega$ in the corresponding frame for the metric $g_\Omega$. Then,

$$\begin{aligned} {R}^\Omega _{kl}=v^2R_{kl}+(n-2)\bigg [vv_{,kl}-\frac{1}{2}g_{kl}|\nabla _gv|^2\bigg ] +g_{kl}\bigg [v\Delta _gv-\frac{n}{2}|\nabla _gv|^2\bigg ]. \end{aligned}$$

By (2.4), we have

$$\begin{aligned} {R}^\Omega _{kl}=v^2R_{kl}+(n-2)\bigg [vv_{,kl}-\frac{1}{2}g_{kl}|\nabla _gv|^2\bigg ] -g_{kl}\bigg [\frac{1}{2(n-1)}v^2S_g+\frac{n}{2}\bigg ], \end{aligned}$$

(2.9)

or

$$\begin{aligned} {R}^\Omega _{kl}=v^2R_{kl}-\frac{1}{2(n-1)}v^2g_{kl}S_g+(n-2)vv_{,kl} -\frac{n-2}{2}g_{kl}|\nabla _gv|^2-\frac{n}{2}g_{kl}. \end{aligned}$$

(2.10)

We emphasize that (2.9) and (2.10) play important roles in the rest of the paper. By (2.5) and (2.7), we obtain

$$\begin{aligned} {R}^\Omega _{kl}=-(n-1)g_{kl}+O(d). \end{aligned}$$

In other words, the Ricci curvatures of conformal metrics $g_\Omega$ are asymptotically equal to $-(n-1)$ near boundary. We note that this holds in arbitrary smooth domains.

If $(M,g)=({\mathbb {R}}^n, g_E)$, then (2.1) and (2.2) reduce to

$$\begin{aligned} \Delta u&= \frac{1}{4}n(n-2) u^{\frac{n+2}{n-2}} \quad \text {in }\,\Omega , \end{aligned}$$

(2.11)

$$\begin{aligned} u&=\infty \quad \text {on }\partial \Omega . \end{aligned}$$

(2.12)

In this case, the function v given by (2.3) satisfies

$$\begin{aligned} v\Delta v=\frac{n}{2}(|\nabla v |^2-1). \end{aligned}$$

(2.13)

Let $g_\Omega$ be the metric given by (2.8) with $g=g_E$, i.e., $g_\Omega =v^{-2}g_E$. Denote by $R^\Omega _{ijij}$ and $R^\Omega _{ij}$ the sectional curvatures and Ricci curvatures of $g_\Omega$ in the orthonormal coordinates of $g_\Omega$, respectively. Then, for $i\ne j$,

$$\begin{aligned} R^\Omega _{ijij}=vv_{ii}+vv_{jj}-|\nabla v|^{2}, \end{aligned}$$

(2.14)

and, for any i, j,

$$\begin{aligned} R^\Omega _{ij}=(n-2)vv_{ij}-\bigg [\frac{n-2}{2}|\nabla v|^{2}+\frac{n}{2}\bigg ]\delta _{ij}. \end{aligned}$$

(2.15)

Hence, for any $i\ne j$,

$$\begin{aligned} R^\Omega _{ijij}=-1+O(d), \end{aligned}$$

and, for any i, j,

$$\begin{aligned} R^\Omega _{ij}=-(n-1)\delta _{ij}+O(d). \end{aligned}$$

Note

$$\begin{aligned} v_i=-\frac{2}{n-2}u^{-\frac{2}{n-2}-1}u_i, \end{aligned}$$

and

$$\begin{aligned} v_{ij}=-\frac{2}{n-2}u^{-\frac{2}{n-2}}\bigg (\frac{u_{ij}}{u}-\frac{n}{n-2}\frac{u_iu_j}{u^2}\bigg ). \end{aligned}$$

(2.16)

We can also express $R^\Omega _{ijij}$ and $R^\Omega _{ij}$ in terms of u.

3 Convex Domains in Euclidean Spaces

In this section, we study Ricci curvatures and sectional curvatures of the complete conformal metrics associated with the Loewner–Nirenberg problem in bounded domains in the Euclidean space. We will prove that the complete conformal metrics in bounded convex domains have negative sectional curvatures. The convexity assumption allows us to apply a convexity theorem by Kennington [10] directly to conformal factors.

Proof of Theorem 1.1

Let u be the solution of (2.11) and (2.12) in $\Omega$ and v be given by (2.3). Then, $g=v^{-2}g_E$ is the complete conformal metric in $\Omega$ with a constant scalar curvature $-n(n-1)$. Denote by $R_{ijij}$ and $R_{ij}$ the sectional curvatures and Ricci curvatures of g in the orthonormal coordinates of g, given by (2.14) and (2.15), respectively. Here, we suppress $\Omega$ from the notations g, $R_{ij}$ and $R_{ijij}$.

By applying the Laplacian operator to (2.13), we get

$$\begin{aligned} v\Delta (\Delta v)+(2-n)\nabla v \nabla (\Delta v)= n|\nabla ^2v|^2-(\Delta v)^2 \ge 0. \end{aligned}$$

First, we assume that the boundary of $\Omega$ is smooth. By (2.6), we have

$$\begin{aligned} \Delta v=-H_{\partial \Omega }-\frac{1}{n-1}H_{\partial \Omega }+O(d)=-\frac{n}{n-1}H_{\partial \Omega }+O(d). \end{aligned}$$

Since $\Omega$ is convex, we have $\Delta v\le 0$ on $\partial \Omega$. By the strong maximum principle, we obtain $\Delta v<0$ in $\Omega$. Therefore, $|\nabla v| <1$ in $\Omega$ by (2.13). Next, we apply [10, Theorems 3.1 and 3.2] in $\Omega$ and conclude that v is concave. In fact, we write (2.13) as

$$\begin{aligned} \Delta v=\frac{n(|\nabla v |^2-1)}{2v}. \end{aligned}$$

Then, we can verify directly the hypothesis (i) of [10, Theorem 3.1] by $|\nabla v| <1$ and the hypothesis (ii) of [10, Theorem 3.2], since $|\nabla v| <1$, $v=0$ on $\partial \Omega$, and $\nabla v$ is the inner unit normal vector on $\partial \Omega$.

For general bounded convex domains, we can obtain the concavity of v by approximations.

Since $v_{ii}\le 0$, by (2.14) and (2.15), we get, for any $i\ne j$,

$$\begin{aligned} R_{ijij}\le 0, \end{aligned}$$

and, for any i,

$$\begin{aligned} R_{ii} \le -\frac{n}{2}. \end{aligned}$$

By (2.16), we also have, for any i,

$$\begin{aligned} \frac{u_{ii}}{u}-\frac{n}{n-2}\frac{u_i^2}{u^2} \ge 0. \end{aligned}$$

(3.1)

Next, we prove that $R_{ijij}$ does not vanish in $\Omega$ for any $i\ne j$. If $R_{ijij}= 0$ at some point $x_0\in \Omega$ for some $i\ne j$, then

$$\begin{aligned} \bigg (\frac{u_{ii}}{u}-\frac{n}{n-2}\frac{u_i^2}{u^2}\bigg )(x_0)=0, \end{aligned}$$

and

$$\begin{aligned} \nabla u(x_0)=0. \end{aligned}$$

In fact, by (2.14) and $vv_{ii}\le 0$, if $R_{ijij}(x_0)=0$, then $\nabla v(x_0)=0$ and thus $\nabla u(x_0)=0$. Hence, $u_{ii}(x_0)=0$. Applying $\partial _i$ twice to the equation (2.11), we get

$$\begin{aligned} \Delta u_{ii}=\frac{1}{4}n(n+2)u^{\frac{n+2}{n-2}}\frac{u_{ii}}{u}+\frac{n(n+2)}{n-2}u^{\frac{n+2}{n-2}}\frac{u_i^2}{u^2}. \end{aligned}$$

Combining with (3.1), we have

$$\begin{aligned} \Delta u_{ii}-\frac{1}{4}(n+2)(n+4)u^{\frac{4}{n-2}}u_{ii} =(n+2)u^{\frac{n+2}{n-2}}\bigg (\frac{n}{n-2}\frac{u_i^2}{u^2}-\frac{u_{ii}}{u}\bigg )\le 0. \end{aligned}$$

By the strong maximum principle, we have $u_{ii}\equiv 0$ in $\Omega$. On the other hand, by $u_{i}(x_0)=0$, we get $u_{i}\equiv 0$ on $\Omega \cap \{x_0+te_i\,|\,t \in {\mathbb {R}}\}$. Therefore, u is constant on $\Omega \cap \{x_0+te_i\,|\,t \in {\mathbb {R}}\}$. This leads to a contradiction. Therefore, we have, for any $i\ne j$,

$$\begin{aligned} R_{ijij}< 0. \end{aligned}$$

Similarly, we have, for any i,

$$\begin{aligned} R_{ii} <-\frac{n}{2}. \end{aligned}$$

This completes the proof. $\square$

We point out that the upper bounds of sectional curvatures and Ricci curvatures in Theorem 1.1 are given by strict inequalities and, in fact, are optimal. To see this, set

$$\begin{aligned} D=\big \{(x_1, \ldots , x_n)\,|-1<x_n<1\big \}\subset {\mathbb {R}}^n. \end{aligned}$$

Let $g_D$ be the complete conformal metric with the constant scalar curvature $-n(n-1)$ in D, and $R^{D}_{ijij}$ and $R^{D}_{ii}$ be the sectional curvatures and Ricci curvatures of $g_D$, respectively. Then, $R^{D}_{ijij}(0)=0$, for $i\ne j$, $i,j \ne n$, and $R^{D}_{ii}(0)=-{n}/{2}$, for $i \ne n$. Set $\Omega _{R}=D\cap B_{R}$. Then, $R^{\Omega _R}_{ijij}(0)\rightarrow 0$, for $i\ne j$, $i,j \ne n$, and $R^{\Omega _R}_{ii}(0)\rightarrow -{n}/{2}$ for $i \ne n$, as $R\rightarrow \infty$.

4 Domains in Compact Manifolds

In this section, we discuss domains in compact Riemannian manifolds without boundary. We construct domains with boundary close to certain sets of low dimension such that the complete conformal metrics with a negative constant scalar curvature have positive Ricci components somewhere. Throughout this section, the Yamabe invariant plays a crucial role. It determines convergence behaviors of conformal factors and, as a consequence, the methods to be employed. In certain cases, we need to employ expansions of the Green’s function, and also the positive mass theorem.

Suppose (M, g) is a compact Riemannian manifold of dimension $n\ge 3$ without boundary. The Yamabe invariant of M is given by

$$\begin{aligned} \lambda (M, [g])=\inf \left \{ \frac{\int _M \big ( |\nabla _g \phi |^2 +\frac{n-2}{4(n-1)} S_g\phi ^2 \big )dV_g }{\big (\int _M \phi^{\frac{2n}{n-2}} dV_g\big )^{\frac{n-2}{n}}} \,\Bigg |\,\phi \in C^{\infty }(M),\phi >0 \right \}. \end{aligned}$$

The conformal Laplacian of (M, g) is given by

$$\begin{aligned} L_g=-\Delta _g +\frac{n-2}{4(n-1)}S_g. \end{aligned}$$

For any function $\psi$ in M, we have

$$\begin{aligned} L_g(u\psi )= u^{\frac{n+2}{n-2}}L_{u^{\frac{4}{n-2}}g}(\psi ). \end{aligned}$$

We first prove a convergence result which plays an important role in this section. According to signs of Yamabe invariants, conformal factors exhibit different convergence behaviors. We note that the maximum principle is applicable to the operator $L_g$ if $S_g\ge 0$.

Lemma 4.1

Suppose (M, g) is a compact Riemannian manifold of dimension $n\ge 3$ without boundary, with a constant scalar curvature $S_g$, and $\Gamma$ is a closed smooth submanifold of dimension d in M, $0\le d\le \frac{n-2}{2}$. Suppose $\Omega _i$ is a sequence of increasing domains with smooth boundary in M which converges to $M \setminus \Gamma$. Let $u_i$ be the solution of (2.1) and (2.2) in $\Omega _i$. Then, for any positive integer m, if $S_g\ge 0$,

$$\begin{aligned} u_i\rightarrow 0\quad {in }C^{m}_{\mathrm {loc}}( M \setminus \Gamma ) { as }i\rightarrow \infty , \end{aligned}$$

(4.1)

and, if $S_g<0$,

$$\begin{aligned} u_i\rightarrow \bigg (\frac{-S_g}{n(n-1)}\bigg )^{\frac{n-2}{4}} \quad {in }C^{m}_{\mathrm {loc}}( M \setminus\Gamma ) { as }i\rightarrow \infty . \end{aligned}$$

(4.2)

Proof

We first consider the case $S_g\ge 0$. By the maximum principle, we have $u_i \ge u_{i+1}$ in $\Omega _i$. It is straightforward to verify, for any m,

$$\begin{aligned} u_i\rightarrow u\quad \text {in }C^{m}_{\mathrm {loc}}( M\setminus\Gamma ) \text { as }i\rightarrow \infty , \end{aligned}$$

where u is a nonnegative solution of (2.1) in $M \setminus \Gamma$. By the second part of [4, p. 398], u is bounded. Let $\rho (x)$ be a positive smooth function in $M \setminus\Gamma$ which equals to $\text {dist}(x, \Gamma )$ in a neighborhood of $\Gamma$ in M. Then,

$$\begin{aligned} \rho (x)\Delta _g \rho (x)\rightarrow n-d-1\quad \text {as }x\rightarrow \Gamma . \end{aligned}$$

Take $\epsilon _0>0$ sufficiently small. Then,

$$\begin{aligned} \Delta _g \rho ^{-\frac{n-2}{2}+\epsilon _0}=\bigg (-\frac{n-2}{2}+\epsilon _0\bigg )\rho ^{-\frac{n+2}{2}+\epsilon _0} \bigg (\rho \Delta _g \rho -\Big (\frac{n}{2}-\epsilon _0\Big )|\nabla _g\rho |^2\bigg ). \end{aligned}$$

Since $d\le \frac{n-2}{2}$, we have $\Delta \rho ^{-\frac{n-2}{2}+\epsilon _0}<0$ near $\Gamma$. For any $\epsilon >0$, we can find $\delta <\epsilon$ sufficiently small such that

$$\begin{aligned} \Delta _g(\delta \rho ^{-\frac{n-2}{2}+\epsilon _0}+\epsilon ) -S_g (\delta \rho ^{-\frac{n-2}{2}+\epsilon _0}+\epsilon ) \le \frac{n-2}{4}(\delta \rho ^{-\frac{n-2}{2}+\epsilon _0}+\epsilon )^{\frac{n+2}{n-2}}\quad \text {in }M\setminus \Gamma . \end{aligned}$$

By the maximum principle, we have

$$\begin{aligned} u \le \delta \rho ^{-\frac{n-2}{2}+\epsilon _0}+\epsilon \quad \text {in }M\setminus\Gamma . \end{aligned}$$

This implies $u\equiv 0$. In conclusion, we obtain (4.1).

We now consider the case $S_g<0$. We first prove $\Delta _g u_i\ge 0$ in $\Omega _i,$ or equivalently

$$\begin{aligned} u_i\ge \bigg (\frac{-S_g}{n(n-1)}\bigg )^{\frac{n-2}{4}} \quad \text {in }\Omega _i. \end{aligned}$$

(4.3)

If (4.3) is violated somewhere, then $u_i$ must assume its minimum at some point $x_0$ in the set

$$\begin{aligned} \Bigg \{x\in \Omega _i :\, \frac{1}{4}n(n-2) u_i^{\frac{n+2}{n-2}} +\frac{n-2}{4(n-1)}S_gu_i< 0 \Bigg \}. \end{aligned}$$

On the other hand, we have $\Delta _g u_i(x_0)\ge 0$, which leads to a contradiction. By taking a difference, we have

$$\begin{aligned} \Delta _g (u_{i+1}-u_{i})=c_i(u_{i+1}-u_{i})\quad \text {in } \Omega _{i}, \end{aligned}$$

where $c_i$ is a nonnegative function in $\Omega _{i}$ by (4.3). The maximum principle implies $u_{i+1}\le u_{i}$ in $\Omega _i$. Then, for any m,

$$\begin{aligned} u_i\rightarrow u\quad \text {in }C^{m}_{\mathrm {loc}}( M \setminus \Gamma ) \text { as }i\rightarrow \infty , \end{aligned}$$

where u is a solution of (2.1) in $M \setminus \Gamma$. By (4.3), we have

$$\begin{aligned} u\ge \bigg (\frac{-S_g}{n(n-1)}\bigg )^{\frac{n-2}{4}}\quad \text {in }M\setminus \Gamma . \end{aligned}$$

(4.4)

For $\epsilon >0$ sufficiently small, let $u^{\epsilon }_i$ be the solution of

$$\begin{aligned} \Delta _g u^{\epsilon }_i&= \epsilon \frac{n(n-2)}{4}( u^{\epsilon }_i)^{\frac{n+2}{n-2}} \quad \text {in }\Omega _i, \end{aligned}$$

(4.5)

$$\begin{aligned} u^{\epsilon }_i&=\infty \quad \text {on }\partial \Omega _i. \end{aligned}$$

(4.6)

The existence of $u^{\epsilon }_i$ can be obtained by the standard method. More specifically, for each integer j, we solve

$$\begin{aligned} \Delta _g u^{\epsilon ,j}_i&= \epsilon \frac{n(n-2)}{4}( u^{\epsilon ,j}_i)^{\frac{n+2}{n-2}} \quad \text {in }\Omega _i, \end{aligned}$$

(4.7)

$$\begin{aligned} u^{\epsilon ,j}_i&=j\quad \text {on }\partial \Omega _i. \end{aligned}$$

(4.8)

By the maximum principle, we have $u^{\epsilon ,j}_i \le u^{\epsilon ,k}_i$ if $j\le k$. For any $x_0 \in \Omega _i$, choose normal coordinates near $x_0$. Then, it is easy to check that

$$\begin{aligned} u_{r, x_0}(x)=\bigg( \frac{2r}{r^{2}-|x|^{2}}\bigg) ^{\frac{n-2}{2}} \end{aligned}$$

is a supersolution of (4.5) when r is sufficiently small, depending on $x_0$. Hence, by the maximum principle, we have for each point x, $u^{\epsilon ,j}_i(x)\le C(x)$, independent of j. Therefore, by standard estimates, $u^{\epsilon ,j}_i$ converges to some $u^{\epsilon }_i$ in $C^{m}_{\mathrm {loc}}(\Omega _i)$ as $j\rightarrow \infty$ for any m, and $u^{\epsilon }_i \in C^{\infty }( \Omega _i)$ is a solution of (4.5)–(4.6).

By the same method as in the proof of the case $S_g\ge 0$, we obtain, for any m,

$$\begin{aligned} u^{\epsilon }_i\rightarrow 0\quad \text {in }C^{m}_{\mathrm {loc}}( M \setminus \Gamma ) \text { as }i\rightarrow \infty . \end{aligned}$$

Next, we can verify

$$\begin{aligned} \Delta _g \bigg [u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \bigg ]&\le \frac{n-2}{4(n-1)}S_g \bigg [u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \bigg ] \\&\qquad +\frac{n(n-2)}{4} \bigg [ u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \bigg ]^{\frac{n+2}{n-2}}. \end{aligned}$$

To prove this, we simply split the last term according to $1=\epsilon +(1-\epsilon )$. Then,

$$\begin{aligned}&\frac{n-2}{4(n-1)}S_g \bigg [u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \bigg ]\\&\qquad +\frac{n(n-2)}{4} \bigg [ u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \bigg ]^{\frac{n+2}{n-2}} -\epsilon \frac{n(n-2)}{4}( u^{\epsilon }_i)^{\frac{n+2}{n-2}}\\&\quad \ge \frac{n(n-2)}{4} \bigg [ u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \bigg ] \cdot \\&\qquad \Bigg \{(1-\epsilon ) \bigg [u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \bigg ]^{\frac{4}{n-2}} +\frac{1}{n(n-1)}S_g\Bigg \}, \end{aligned}$$

which is nonnegative. By the maximum principle, we have

$$\begin{aligned} u_i \le u^{\epsilon }_i+ \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}} \quad \text {in }\Omega _i, \end{aligned}$$

where we can verify the boundary condition by the polyhomogeneous expansions of $u_i$ and $u^{\epsilon }_i$. Therefore, we have

$$\begin{aligned} u\le \bigg (\frac{-S_g}{(1-\epsilon )n(n-1)}\bigg )^{\frac{n-2}{4}}\quad \text {in }M\setminus \Gamma . \end{aligned}$$

(4.9)

This holds for any $\epsilon \in (0,1)$. Combining (4.4) and (4.9), we obtain

$$\begin{aligned} u=\bigg (\frac{-S_g}{n(n-1)}\bigg )^{\frac{n-2}{4}}. \end{aligned}$$

In conclusion, we obtain (4.2). $\square$

A similar result holds if the scalar curvature has a fixed sign, not necessarily constant.

Now, we study the case that the boundary is close to a closed smooth submanifold of low dimension. The result below holds for all compact manifolds without boundary, but different signs of the Yamabe invariants require different methods, mostly due to the different convergence behaviors as in Lemma 4.1.

Theorem 4.2

Suppose (M, g) is a compact Riemannian manifold of dimension $n\ge 3$ without boundary and $\Gamma$ is a closed smooth submanifold of dimension d in M, $1\le d\le \frac{n-2}{2}$. Suppose $\Omega _i$ is a sequence of increasing domains with smooth boundary in M which converges to $M\setminus \Gamma$ and $g_i$ is the complete conformal metric in $\Omega _i$ with the scalar curvature $-n(n-1)$. Then, for sufficiently large i, $g_{i}$ has a positive Ricci curvature component somewhere in $\Omega _{i}$. Moreover, the maximal Ricci curvature of $g_i$ in $\Omega _{i}$ diverges to $\infty$ as $i\rightarrow \infty$.

Proof

Let $u_i$ be the solution of (2.1) and (2.2) in $\Omega _i$ and set $v_i=u_i^{-\frac{2}{n-2}}$. Then,

$$\begin{aligned} g_{i}=u_i^{\frac{4}{n-2}}g=v_i^{-2}g. \end{aligned}$$

By the solution of the Yamabe problem, we can assume the scalar curvature $S_g$ of M is the constant $\lambda (M,[g])$. Since M is compact, we can take $\Lambda >0$ such that

$$\begin{aligned} |R_{ij}|\le \Lambda g_{ij}. \end{aligned}$$

We now discuss two cases according to the sign of $S_g$.

Case 1. We first consider the case $S_g\ge 0$. By Lemma 4.1, for any m,

$$\begin{aligned} u_i\rightarrow 0\quad \text {in }C^{m}_{\mathrm {loc}}( M \setminus \Gamma ) \text { as }i\rightarrow \infty , \end{aligned}$$

and hence

$$\begin{aligned} v_i \text { diverges to } \infty \text { locally uniformly in } M \setminus \Gamma \text { as } i\rightarrow \infty . \end{aligned}$$

We now consider two subcases.

Case 1.1. $\Gamma$ is not totally geodesic. For any $\epsilon >0$, there exist two points $p,q \in \Gamma$, such that the length of the shortest geodesic $\sigma _{pq}$ connecting p and q is less than $\epsilon$ and $\sigma _{pq}\cap \Gamma =\{p, q\}$. When $\epsilon$ is sufficiently small, we can assume q is located in a small neighborhood of p covered by normal coordinates. Without loss of generality, we assume $p=0$ and $q=Le_n$.

For i large, set $p_i={\widehat{t}}_ie_n$ and $q_i={\widetilde{t}}_ie_n$, where

$$\begin{aligned} {\widehat{t}}_i&=\min \{t'\in [0,{L}/{2}] \,|\,te_n \in \Omega _i , \text { for any } t\in (t',{L}/{2}]\},\\ {\widetilde{t}}_i&= \max \{t'\in [{L}/{2},L]\,|\,te_n \in \Omega _i , \text { for any } t\in [{L}/{2},t')\}. \end{aligned}$$

Then, $p_i, q_i\in \partial \Omega _i$. By the convergence of $\Omega _i$ to $M\setminus \Gamma$, we have

$$\begin{aligned} p_i\rightarrow p,\quad q_i\rightarrow q. \end{aligned}$$

By the polyhomogeneous expansions of $v_i$, we have

$$\begin{aligned} |\partial _n v_i(p_i)|\le C_i \quad \text {and}\quad |\partial _n v_i(q_i)|\le C_i, \end{aligned}$$

where $C_i$ is some positive constant which converges to 1 as $i\rightarrow \infty$ and $\epsilon \rightarrow 0$.

Since $v_i(Le_n/2)\rightarrow \infty$ as $i\rightarrow \infty$, for i large, we can take $t_i \in ({\widehat{t}}_i,{\widetilde{t}}_i)$ such that, for any $t\in ({\widehat{t}}_i,{\widetilde{t}}_i)$,

$$\begin{aligned} \partial _nv_i(te_n)\le \partial _nv_i(t_ie_n). \end{aligned}$$

Then,

$$\begin{aligned} \partial _nv_i(t_ie_n)>\frac{v_i(\frac{L}{2}e_n)-0}{\frac{L}{2}}\ge \frac{2}{L}v_i\left( \frac{L}{2}e_n\right) , \end{aligned}$$

and

$$\begin{aligned} \partial _{nn}v_i( t_ie_n)=0. \end{aligned}$$

We also have

$$\begin{aligned} |v_i( t_ie_n)|\le \frac{L}{2}\partial _nv_i(t_ie_n). \end{aligned}$$

(4.10)

Denote by $R^i_{nn}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i \frac{\partial }{\partial x^n}$ with respect to the metric $g_i$. By (2.10), we have, at $t_ie_n$,

$$\begin{aligned} R^i_{nn}\le v_i^2|R_{nn}|-\frac{n-2}{2}(\partial _nv_i)^2\le \bigg [\frac{L^2}{4}|R_{nn}|-\frac{n-2}{2}\bigg ](\partial _nv_i)^2\rightarrow -\infty , \end{aligned}$$

if L is sufficiently small. Hence, some component of the Ricci curvature of $g_i$ at the point $t_ie_n$ diverges to $\infty$ as $i\rightarrow \infty$.

Case 1.2. $\Gamma$ is totally geodesic. Fix a point $x_0 \in \Gamma$ and choose normal coordinates near $x_0$ such that $x_{0}=0$ and $\Gamma$ near $x_0$ is given by $x_i=0$, $i=1,\ldots ,n-d$. Consider the curve $\sigma$ given by

$$\begin{aligned} \sigma (t)=(\sqrt{R^2-t^2}-\sqrt{R^2-\epsilon ^2}, 0, \ldots , 0, t)\quad \text {for }t \in [-\epsilon ,\epsilon ], \end{aligned}$$

where R is some sufficiently large constant and $\epsilon$ is some sufficiently small constant such that $\sigma \cap \Gamma =\{ \sigma (-\epsilon ), \sigma (\epsilon )\}$.

For i large, set $p_i=\sigma ({\widehat{t}}_i)$ and $q_i=\sigma ( {\widetilde{t}}_i)$, where

$$\begin{aligned} {\widehat{t}}_i&=\min \{t'\in [-\epsilon ,0]\,|\,\sigma (t) \in \Omega _i, \text { for any } t\in ( t',0]\},\\ {\widetilde{t}}_i&=\max \{t'\in [0,\epsilon ]\,|\,\sigma (t) \in \Omega _i, \text { for any } t\in [0,t')\}. \end{aligned}$$

Then, $p_i, q_i\in \partial \Omega _i$ and

$$\begin{aligned} p_i\rightarrow \sigma (-\epsilon ), \quad q_i\rightarrow \sigma (\epsilon ). \end{aligned}$$

By the polyhomogeneous expansion of $v_i$, we have

$$\begin{aligned} |\partial _{n} v_i(\sigma ({\widehat{t}}_i))|\le C_i\quad \text {and}\quad |\partial _{n} v_i(\sigma ({\widetilde{t}}_i))|\le C, \end{aligned}$$

where C is some positive bounded constant independent of i.

Consider the single variable function $(v_i\circ \sigma )(t)$. Since $(v_i\circ \sigma )(0)\rightarrow \infty$ as $i\rightarrow \infty$, for i large, we can take $t_i \in ({\widehat{t}}_i,{\widetilde{t}}_i)$ such that, for any $t\in ({\widehat{t}}_i,{\widetilde{t}}_i)$,

$$\begin{aligned} \partial _{t}(v_i\circ \sigma )(t)\le \partial _{t} (v_i\circ \sigma )(t_i). \end{aligned}$$

Then,

$$\begin{aligned} \partial _{t} (v_i\circ \sigma )(t_i)>\frac{1}{\epsilon } (v_i\circ \sigma )(0), \end{aligned}$$

and

$$\begin{aligned} \partial _{tt} (v_i\circ \sigma )(t_i)=0. \end{aligned}$$

(4.11)

We also have

$$\begin{aligned} |(v_i\circ \sigma )(t_i)|\le \epsilon \partial _{t} (v_i\circ \sigma )(t_i). \end{aligned}$$

(4.12)

Note that

$$\begin{aligned} \partial _{t}(v_i\circ \sigma )(t_i)= (\partial _n v_i)(\sigma (t_i))-\frac{t_i}{\sqrt{R^2-t_i^2}}(\partial _1 v_i)(\sigma (t_i)). \end{aligned}$$

Set

$$\begin{aligned} \nu _i=\frac{\partial }{\partial x_n}-\frac{t_i}{\sqrt{R^2-t_i^2}}\frac{\partial }{\partial x_1}. \end{aligned}$$

By (4.11), we have

$$\begin{aligned} ( \partial _{\nu _i\nu _i}v_i)(\sigma (t_i)) =\Bigg (-\frac{1}{\sqrt{R^2-t_i^2}}+\frac{t_i^2}{(R^2-t_i^2)^{\frac{3}{2}}}\Bigg )\partial _1 v_i(\sigma (t_i)). \end{aligned}$$

Hence, $( \partial _{\nu _i\nu _i}v_i)(\sigma (t_i))$ is sufficiently small compared with $( |\nabla v_i|)(\sigma (t_i))$, for R sufficiently large and $\epsilon$ sufficiently small. Write $g_{\nu _i\nu _i}=g (\nu _i, \nu _i)$ and denote by $R^i_{\nu _i\nu _i}$ the Ricci curvature of $g_i$ acting on the unit vector $\frac{v_i\nu _i}{\sqrt{g_{\nu _i\nu _i}}}$ with respect to the metric $g_i$. Similarly as in Case 1.1, we can verify at the point $\sigma (t_i)$, $R^i_{\nu _i\nu _i}$ diverges to $-\infty$ as $i\rightarrow \infty$. Hence, some component of the Ricci curvature of $g_i$ at the point $\sigma (t_i)$ diverges to $\infty$ as $i\rightarrow \infty$.

Case 2. We now consider the case $S_g<0$. By Lemma 4.1, for any m,

$$\begin{aligned} u_i\rightarrow \bigg (\frac{-S_g}{n(n-1)}\bigg )^{\frac{n-2}{4}} \quad \text {in }C^{m}_{\mathrm {loc}}( M \setminus\Gamma ) \text { as }i\rightarrow \infty , \end{aligned}$$

and hence

$$\begin{aligned} v_i \rightarrow \bigg (\frac{-S_g }{n(n-1)}\bigg )^{-\frac{1}{2}} \quad \text {in }C^{m}_{\mathrm {loc}}( M \setminus \Gamma ) \text { as }i\rightarrow \infty . \end{aligned}$$

(4.13)

Fix a point $x_0 \in \Gamma$ and choose normal coordinates in a small neighborhood of $x_{0}$ such that $x_{0}=0$ and $x_n$-axis is a normal geodesic of $\Gamma$ near $x_0$. Take $\epsilon >0$ sufficiently small. For i large, set $p_i=t_ie_n$, where

$$\begin{aligned} t_i=\min \{t'\in [0,\epsilon ]\,|\,te_n \in \Omega _i, \text { for any }t\in (t', \epsilon ]\}. \end{aligned}$$

Then, $p_i\in \partial \Omega _i$ and

$$\begin{aligned} p_i\rightarrow 0. \end{aligned}$$

By the polyhomogeneous expansion of $v_i$, we have

$$\begin{aligned} |\partial _n v_i(p_i)|\le C_i, \end{aligned}$$

where $C_i$ is some positive constant which converges to 1 as $i\rightarrow \infty$. By (4.13),

$$\begin{aligned} \frac{\partial v_i}{\partial x_n}(\epsilon e_n )\rightarrow 0\quad \text {as }i\rightarrow \infty . \end{aligned}$$

For i large, we take ${\widetilde{t}}_i \in (t_i,\epsilon )$ such that, for any $t\in (t_i,\epsilon )$,

$$\begin{aligned} \partial _nv_i(te_n)\le \partial _nv_i({\widetilde{t}}_ie_n). \end{aligned}$$

Then,

$$\begin{aligned} \partial _nv_i({\widetilde{t}}_ie_n)>\frac{v_i(\epsilon e_n)-0}{\epsilon -t_i}> \frac{1}{2}\epsilon ^{-1}\bigg (\frac{-S_g }{n(n-1)}\bigg )^{-\frac{1}{2}}, \end{aligned}$$

and

$$\begin{aligned} \partial _{nn}v_i(\widetilde{ t}_ie_n)=0. \end{aligned}$$

Denote by $R^i_{nn}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i \frac{\partial }{\partial x^n}$ with respect to the metric $g_i$. Similarly, by (2.10) at the point ${\widetilde{t}}_ie_n$, $R_{nn}^i\le -C\epsilon ^{-2}$, for all large i, for some positive constant C independent of i and $\epsilon$. By choosing appropriate $\epsilon$, we have the desired result. $\square$

Next, we discuss the case that the boundary is close to a point $x_0$. The proof of the next result is rather delicate if the Yamabe invariant is positive, in which case expansions of the Green’s function play an essential role. We need to employ the positive mass theorem if the manifold has a dimension 3, 4, or 5, or is locally conformally flat. In the case that $n\ge 6$ and M is not conformally flat in a neighborhood of $x_0$, we need to analyze Weyl tensors and distinguish two cases $W(x_0) \ne 0$ and $W(x_0)=0$. The proof for the case $W(x_0)=0$ is quite delicate. It is worth emphasizing that the Weyl tensor can be zero at $x_0$ even if M is not conformally flat in a neighborhood of $x_0$. Different vanishing orders of W at $x_0$ requires different methods. In fact, we also need to employ the positive mass theorem if the Weyl tensor vanishes at $x_0$ up to a sufficiently high order.

Theorem 4.3

Let (M, g) be a compact Riemannian manifold of dimension $n\ge 3$ without boundary, with $\lambda (M,[g])< \lambda (S^n,[g_{S^n}])$, where $S^n$ is the sphere with its standard metric $g_{S^n}$, and let $x_0$ be a point in M. Suppose that $\Omega _i$ is a sequence of increasing domains with smooth boundary in M which converges to $M\setminus\{x_0\}$ and $g_{i}$ is the complete conformal metric in $\Omega _i$ with the scalar curvature $-n(n-1)$. Then, for i sufficiently large, $g_i$ has a positive Ricci curvature component somewhere in $\Omega _{i}$. Moreover, the maximal Ricci curvature of $g_i$ in $\Omega _{i}$ diverges to $\infty$ as $i\rightarrow \infty$.

Proof

Let $u_i$ be the solution of (2.1) and (2.2) in $\Omega _i$ and set $v_i=u_i^{-\frac{2}{n-2}}$. Then,

$$\begin{aligned} g_{i}=u_i^{\frac{4}{n-2}}g=v_i^{-2}g. \end{aligned}$$

We consider several cases according to the sign of the Yamabe invariant $\lambda (M,[g])$.

Case 1. We first consider $\lambda (M,[g])<0$. We point out that the proof of Case 2 of Theorem 4.2 can be adapted to yield the conclusion.

Case 2. Next, we consider $\lambda (M,[g])=0$. As in the proof of Theorem 4.2, we assume the scalar curvature of M is 0 and

$$\begin{aligned} |R_{ij}|\le \Lambda g_{ij}. \end{aligned}$$

Let $\delta$ be some small positive constant such that $\Lambda \delta <{1}/{10}$ and there exist normal coordinates in $B_{\delta }(x_0 )$.

Take a sufficiently small $r>0$ with $r\le \delta$. Since $\Omega _i\rightarrow M \setminus \{x_0\}$, we have $M \setminus B_{r}(x_0 )\subset \subset \Omega _i$ for i large. For such i, by the Harnack inequality, we have

$$\begin{aligned} \max u_i \le C \min u_i \quad \text {in }M \setminus B_{r}(x_0 ), \end{aligned}$$

where C is some positive constant depending only on n, M and r. Then for i sufficiently large, by (4.1), we have

$$\begin{aligned} |\nabla _g u_i| \le C\Big( u_i+ u_i^{\frac{n+2}{n-2}}\Big) \le Cu_i \quad \text {in }M \setminus B_{r}(x_0 ). \end{aligned}$$

(4.14)

We denote by $m_i$ the minimum of $u_i$ in $\Omega _i$. With the definition of $v_i$, we have, for i sufficiently large,

$$\begin{aligned} |\nabla _g v_i| \le C v_i \le C m_i^{-\frac{2}{n-2}} \quad \text {in }M \setminus B_{r}(x_0 ), \end{aligned}$$

(4.15)

where C is some positive constant depending only on n, M and $\delta$. Set

$$\begin{aligned} A_i=\{x\in \Omega _i\,|\,u_i(x)<2m_i\}. \end{aligned}$$

Then, for any fixed $r>0$ with $r\le \delta$ and any i sufficiently large, we have $A_i\cap B_{r}(x_0 ) \ne \emptyset .$ Otherwise, by the maximum principle, we have

$$\begin{aligned} u_i(x) \ge 2m_i-Cm_i^{\frac{n+2}{n-2}}\quad \text {in }A_i, \end{aligned}$$

where C is some constant depending only on n, M and r. Hence,

$$\begin{aligned} m_i \ge 2m_i-Cm_i^{\frac{n+2}{n-2}}. \end{aligned}$$

Note that $m_i\rightarrow 0$ as $i\rightarrow \infty$, which leads to a contradiction.

By $v_i=0$ on $\partial \Omega _i$, we have, for any fixed $r>0$ with $r\le \delta$ and for any i sufficiently large,

$$\begin{aligned} |\nabla _g v_i| \ge \frac{1}{r}(2m_i)^{-\frac{2}{n-2}}\quad \text {somewhere in }\Omega _i\cap B_{r}(x_0 ). \end{aligned}$$

Therefore, for i sufficiently large, $|\nabla _g v_i|$ must assume its maximum at $p_i \in \Omega _i\cap B_{\delta }(x_0 )$. Write $\nu _i=\frac{\nabla _g v_i}{|\nabla _g v_i|}$ and denote by $R^i_{\nu _i\nu _i}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\nu _i$ with respect to the metric $g_i$. Then, we can proceed as in the proof of Theorem 4.2 to verify at the point $p_i$, $R^i_{\nu _i\nu _i}$ diverges to $-\infty$ as $i\rightarrow \infty$. Hence, some component of the Ricci curvature of $g_i$ at the point $p_i$ diverges to $\infty$ as $i\rightarrow \infty$.

Case 3. We now consider the case $\lambda (M,[g])>0$. In this case, there exists $G_{x_0}\in C^{\infty }(M \setminus \{x_0\})$, the Green’s function for the conformal Laplacian $L_g$, such that

$$\begin{aligned} L_gG_{x_0}=(n-2)\omega _{n-1}\delta _{x_0},\quad G_{x_0}>0, \end{aligned}$$

where $\omega _{n-1}$ is the volume of $S^{n-1}$. Up to a conformal factor, we can assume (M, g) has conformal normal coordinates near $x_0$. See [13, p. 69] or [24, Chapter 5]. We can perform a conformal blow up at $x_0$ to obtain an asymptotic flat and scalar flat manifold using $G_{x_0}$. Specifically, if we define the metric ${\widetilde{g}} = G_{x_0}^{\frac{4}{n-2}}g$ on ${\widetilde{M}} =M\setminus \{x_0\}$, then, $({\widetilde{M}}, {\widetilde{g}})$ is an asymptotically flat and scalar flat manifold, and ${\widetilde{g}}$ has an asymptotic expansion near infinity. See [13, pp. 64–65] or [24, Chapter 5].

Set ${\widetilde{u}}_i=u_i/ G_{x_0}$. Then, ${\widetilde{u}}_i$ satisfies

$$\begin{aligned} \Delta _{{\widetilde{g}}} {\widetilde{u}}_{i}&= \frac{1}{4}n(n-2) {\widetilde{u}}_{i}^{\frac{n+2}{n-2}} \quad \text {in }\,\Omega _i, \end{aligned}$$

(4.16)

$$\begin{aligned} {\widetilde{u}}_{i}&=\infty \quad \text {on }\partial \Omega _i, \end{aligned}$$

(4.17)

and for any m,

$$\begin{aligned} {\widetilde{u}}_i\rightarrow 0\quad \text {in }C^{m}_{\mathrm {loc}}( M \setminus \{x_0\}) \text { as }i\rightarrow \infty . \end{aligned}$$

(4.18)

Fix a point $p_0\in M \setminus \{x_0\}$. Then, $p_0 \in \Omega _i$, for i sufficiently large. Set ${\widetilde{w}}_i={{\widetilde{u}}_i}/{{\widetilde{u}}_i(p_0)}$. Then, ${\widetilde{w}}_i(p_0)=1$, and ${\widetilde{w}}_i$ satisfies

$$\begin{aligned} \Delta _{{\widetilde{g}}} {\widetilde{w}}_{i}&= \frac{1}{4}n(n-2)u_i(p_0)^{\frac{4}{n-2}} {\widetilde{w}}_{i}^{\frac{n+2}{n-2}} \quad \text {in }\,\Omega _i, \end{aligned}$$

(4.19)

$$\begin{aligned} {\widetilde{w}}_{i}&=\infty \quad \text {on }\partial \Omega _i. \end{aligned}$$

(4.20)

By interior estimates, there exists a positive function ${\widetilde{w}}\in {\widetilde{M}}$ such that, for any m,

$$\begin{aligned} {\widetilde{w}}_i\rightarrow {\widetilde{w}}\quad \text {in }C^{m}_{\mathrm {loc}}( {\widetilde{M}}) \text { as }i\rightarrow \infty , \end{aligned}$$

and

$$\begin{aligned} \Delta _{{\widetilde{g}}} {\widetilde{w}}=0 \quad \text {in } {\widetilde{M}}. \end{aligned}$$

(4.21)

Hence,

$$\begin{aligned} L_g( G_{x_0}{\widetilde{w}})=0 \quad \text {in } M \setminus \{x_0\}. \end{aligned}$$

(4.22)

By the expansion of $G_{x_0}$ near $x_0$ and [14, Proposition 9.1], we conclude that ${\widetilde{w}}$ converges to some constant as $x\rightarrow x_0$. Therefore, ${\widetilde{w}}\equiv 1$ in ${\widetilde{M}}$. Hence, for any m,

$$\begin{aligned} \frac{u_i}{{\widetilde{u}}_i (p_0) G_{x_0}}\rightarrow 1 \quad \text {in }C^{m}_{\mathrm {loc}}( M\setminus\{x_0\}) \text { as }i\rightarrow \infty . \end{aligned}$$

(4.23)

In the following, we always discuss in the conformal normal coordinates near $x_0$. Set

$$\begin{aligned} v_i= u_i^{-\frac{2}{n-2}}= ({\widetilde{u}}_i (p_0))^{-\frac{2}{n-2}} \bigg (\frac{u_i}{{\widetilde{u}}_i (p_0) G_{x_0}}\bigg )^{-\frac{2}{n-2}}G_{x_0}^{-\frac{2}{n-2}}. \end{aligned}$$

We will fix a direction appropriately, which we call $x_1$. Denote by $R^i_{11}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\frac{\partial }{\partial x_1}$ with respect to the metric $g_i$. To study $R^i_{11}$ given by (2.9), we need to analyze the expansion of $G_{x_0}^{-\frac{2}{n-2}}.$ See [13] or [24] for details.

Now we discuss several cases.

Case 3.1. $n=3,4,5$, or M is conformally flat in a neighborhood of $x_0$. In this case, we have

$$\begin{aligned} G_{x_0}=r^{2-n}+A+O(r), \end{aligned}$$

where A is a constant. Since $\lambda (M,[g])< \lambda (S^n,[g_{S^n}])$, we have $A>0$ when $3\le n \le 7$, or M is locally conformally flat, or M is spin. We also have $A>0$ when M is just conformally flat in a neighborhood of $x_0$ under the assumption that the positive mass theorem holds. Then,

$$\begin{aligned}&G_{x_0}^{-\frac{2}{n-2}}=r^2-\frac{2}{n-2}Ar^n+O( r^{n+1}),\\ &\partial _rG_{x_0}^{-\frac{2}{n-2}}=2r-\frac{2n}{n-2}Ar^{n-1}+O( r^{n}),\\ &\partial _{rr}G_{x_0}^{-\frac{2}{n-2}}=2-\frac{2n(n-1)}{n-2}Ar^{n-2}+O( r^{n-1}), \end{aligned}$$

and hence

$$\begin{aligned} G_{x_0}^{-\frac{2}{n-2}} \partial _{rr}G_{x_0}^{-\frac{2}{n-2}} -\frac{1}{2}\Big( \partial _r G_{x_0}^{-\frac{2}{n-2}}\Big)^2 =-2(n-1)Ar^n+ O( r^{n+1}). \end{aligned}$$

For $n=3,4,5$, by [13, p. 61], $R_{11}(x_0)=0$, $R_{11,1}(x_0)=0$ and $R_{11,11}(x_0)\le 0$, we have

$$\begin{aligned} R_{11}\le C|x_1|^3\quad \text {on the } x_1\text {-axis near }x_0=0. \end{aligned}$$

We also have $S_{g}(x_0)=0$ and $S_{g,1}(x_0)=0$, and hence

$$\begin{aligned} |S_g |\le Cx_1^2\quad \text {on the } x_1\text {-axis near }x_0=0. \end{aligned}$$

Take any $x_1>0$ small. Then, at the point $x_1 e_1$,

$$\begin{aligned} v_i^2R_{11}\le C({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}x_1^7, \end{aligned}$$

(4.24)

and

$$\begin{aligned} v_i^2|S_g|\le C({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}x_1^6. \end{aligned}$$

(4.25)

For i large, by (2.9), we have, at the point $x_1 e_1$,

$$\begin{aligned} R^i_{11}\le ({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}\big [-2(n-1)(n-2)Ax_1^n+Cx_1^6+o(1)\big ], \end{aligned}$$

(4.26)

where o(1) denotes terms converging to zero as $i\rightarrow \infty$, uniformly for small $x_1$ away from 0. The dominant term in (4.26) is the $x_1^n$-term, with a negative coefficient. Hence, the expression inside the bracket in (4.26) is strictly less than 0, for a fixed small $x_1\ne 0$ and i large. Therefore, at the point $x_1e_1$, $R^i_{11}$ diverges to $-\infty$ as $i\rightarrow \infty$. Hence, some component of the Ricci curvature of $g_i$ at the point $x_1e_1$ diverges to $\infty$ as $i\rightarrow \infty$.

If M is conformally flat in a neighborhood of $x_0$, then $R_{11}=0$ and $S_{g}=0$ on the $x_1$-axis and near $x_0=0$. The $x_1^6$-term in (4.26) is absent. Similarly, at the point $x_1e_1$ for $x_1>0$ sufficiently small, $R^i_{11}$ diverges to $-\infty$ as $i\rightarrow \infty$.

If we denote by $R^i_{rr}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\frac{\partial }{\partial r}$ with respect to the metric $g_i$, then we conclude similarly that $R^i_{rr}$ at x diverges to $-\infty$ as $i\rightarrow \infty$, for some x sufficiently close to $x_0$.

Case 3.2. $n=6$ and M is not conformally flat in a neighborhood of $x_0$. In this case,

$$\begin{aligned} G_{x_0}(x)=r^{2-n}-\frac{n-2}{1152(n-1)} |W(x_0)|^2\log r -\frac{1}{96}S_{g,ij}(x_0)\frac{x^{i}x^{j}}{r^{2}} +P(x)\log r +\alpha (x), \end{aligned}$$

where W is the Weyl tensor, P(x) is a polynomial with $P(0)=0$, and $\alpha$ is a $C^{2,\mu }$-function. We note that $W_{ijkl}$ is given by

$$\begin{aligned} \begin{aligned} W_{ijkl}&=R_{ijkl}-\frac{1}{n-2}\big (R_{ik}g_{jl}-R_{il}g_{jk} +R_{jl}g_{ik}-R_{jk}g_{il}\big )\\&\quad\ +\frac{S_{g}}{(n-1)(n-2)}\big (g_{ik}g_{jl}-g_{il}g_{jk}\big ). \end{aligned} \end{aligned}$$

(4.27)

Case 3.2.1. If $W(x_0) \ne 0$, then,

$$\begin{aligned} &G_{x_0}^{-\frac{2}{n-2}}=r^2+\frac{1}{2880}|W(x_0)|^2r^6\log r + O(r^7\log r) ,\\ &\partial _rG_{x_0}^{-\frac{2}{n-2}}=2r+\frac{1}{480}|W(x_0)|^2r^5\log r+O( r^{5}),\\ &\partial _{rr}G_{x_0}^{-\frac{2}{n-2}}=2+\frac{1}{96}|W(x_0)|^2r^4\log r+O( r^{4}), \end{aligned}$$

and hence

$$\begin{aligned} G_{x_0}^{-\frac{2}{n-2}} \partial _{rr}G_{x_0}^{-\frac{2}{n-2}} -\frac{1}{2}\Big( \partial _r G_{x_0}^{-\frac{2}{n-2}}\Big)^2 =\frac{1}{144}|W(x_0)|^2r^6 \log r+ O( r^{6}). \end{aligned}$$

Take any $x_1>0$ small. Then, at the point $x_1 e_1$, (4.24) and (4.25) still hold. For i large, instead of (4.26), we have, at the point $x_1 e_1$,

$$\begin{aligned} R^i_{11}\le ({\widetilde{u}}_i(p_0))^{-1}\bigg [\frac{1}{36}|W(x_0)|^2x_1^6 \log x_1+Cx_1^6+o(1)\bigg ]. \end{aligned}$$

Similarly as in Case 3.1, at the point $x_1e_1$ for $x_1>0$ sufficiently small, $R^i_{11}$ diverges to $-\infty$ as $i\rightarrow \infty$.

Similarly, $R^i_{rr}$ at x diverges to $-\infty$ as $i\rightarrow \infty$, for some x sufficiently close to $x_0$.

Case 3.2.2. We now consider the case $W(x_0)=0$. By (4.27), we have $R_{ijkl}(x_0)=0$. Hence, $({\widetilde{M}}, {\widetilde{g}})$ is asymptotically flat of order 3. Using the spherical coordinates, we set

$$\begin{aligned} \phi _4(\theta )=\frac{1}{96}S_{g,ij}(x_0)\frac{x^{i}x^{j}}{r^{2}}, \end{aligned}$$

and denote by $r^2g_2(\theta )$ the degree two part of the Taylor expansion of $S_{g}$ at $x_0$. Since

$$\begin{aligned} \sum _{i=1}^{n}S_{g,ii}(x_0)=-\frac{1}{6}|W|^2(x_0)=0, \end{aligned}$$

then,

$$\begin{aligned} \int _{S^{n-1}}g_2(\theta )d\theta =0. \end{aligned}$$

By the positive mass theorem, see [13, pp. 79–80], we have

$$\begin{aligned} \int _{S^{n-1}}\big (\phi _4(\theta )+\alpha (0)\big )d\theta >0. \end{aligned}$$

Along a radial geodesic $\{(r,\theta ):\,0\le r\le \delta \}$, for a small constant $\delta$, we have

$$\begin{aligned} &G_{x_0}^{-\frac{2}{n-2}}=r^2-\frac{1}{2}\big (\phi _4(\theta )+\alpha (0) \big ) r^6 +o( r^{6}),\\ &\partial _rG_{x_0}^{-\frac{2}{n-2}}=2r-3\big (\phi _4(\theta )+\alpha (0) \big ) r^{5}+o( r^{5}),\\ &\partial _{rr}G_{x_0}^{-\frac{2}{n-2}}=2-15\big (\phi _4(\theta )+\alpha (0) \big ) r^{4}+o( r^{4}), \end{aligned}$$

and hence

$$\begin{aligned} G_{x_0}^{-\frac{2}{n-2}} \partial _{rr}G_{x_0}^{-\frac{2}{n-2}} -\frac{1}{2}\Big( \partial _r G_{x_0}^{-\frac{2}{n-2}}\Big)^2 =-10\big (\phi _4(\theta )+\alpha (0) \big ) r^n+ o( r^{n}).\end{aligned}$$

By [13, p. 61], along the radial geodesic $(\cdot ,\theta )$, $\partial _r^3R_{rr}(x_0)=0$ and $\partial _r^4R_{rr}(x_0)\le 0$. Therefore, for i large, by (2.9), we have, along the radial geodesic $(\cdot ,\theta )$,

$$\begin{aligned} R^i_{rr}|_{(r,\theta )}\le ({\widetilde{u}}_i(p_0))^{-1}\bigg [-\frac{1}{10}g_2(\theta )r^6 -40\big (\phi _4(\theta )+\alpha (0) \big )r^6 +o(r^6)+o(1)\bigg ], \end{aligned}$$

where o(1) denotes terms converging to zero as $i\rightarrow \infty$, uniformly for small x away from 0. Hence,

$$\begin{aligned} \int _{S^{n-1}}R^i_{rr}d\theta \le ({\widetilde{u}}_i(p_0))^{-1} \bigg [-40r^6\int _{S^{n-1}}\big (\phi _4(\theta )+\alpha (0) \big )d\theta +o(r^6)+o(1)\bigg ]. \end{aligned}$$

Therefore, we can find $\theta _0\in S^{n-1}$ that $R^i_{rr}$ at $(r,\theta _0)$ diverges to $-\infty$ as $i\rightarrow \infty$, for some r sufficiently small.

Case 3.3. $n\ge 7$ and M is not conformally flat in a neighborhood of $x_0$. In this case,

$$\begin{aligned} G_{x_0}(x)=r^{2-n}\Bigg [1+\sum _{i=4}^{n}\psi _{i}\Bigg ]+c\log r+P(x)\log r+\alpha (x), \end{aligned}$$

where $\psi _{i}$ is a homogeneous polynomial of degree i, c is a constant, P(x) is a polynomial with $P(0)=0$, and $\alpha$ is a $C^{2,\mu }$-function. We note that $c=0$ and $P\equiv 0$ if n is odd. Moreover,

$$\begin{aligned} \psi _4(x)=\frac{n-2}{48(n-1)(n-4)}\bigg (\frac{r^4}{12(n-6)} |W(x_0)|^2-S_{g,ij}(x_0)x^i x^jr^2\bigg ), \end{aligned}$$

where W is the Weyl tensor.

Case 3.3.1. First, we consider the case $|W(x_0)| \ne 0$. Note that $S_{g}(x_0)=0$, $\nabla _gS_{g}(x_0)=0$, and

$$\begin{aligned} \Delta _{g}S_{g}(x_0)=-\frac{1}{6}|W(x_0)|^2. \end{aligned}$$

Without loss of generality, we assume $S_{g,11}(x_0)<0$. Take any $x_1>0$ small. Then, at the point $x_1 e_1$, (4.24) still holds. Set

$$\begin{aligned} A=\frac{n-2}{48(n-1)(n-4)}\bigg [\frac{1}{12(n-6)} |W(x_0)|^2 -S_{g,11}(x_0)\bigg ]. \end{aligned}$$

Then, on the positive $x_1$-axis near $x_0=0$, we have

$$\begin{aligned} &G_{x_0}^{-\frac{2}{n-2}}=x_1^2-\frac{2}{n-2}Ax_1^6 +O(x_1^7) ,\\ &\partial _{x_1}G_{x_0}^{-\frac{2}{n-2}}=2x_1-\frac{12}{n-2}Ax_1^5 +O(x_1^6),\\ &\partial _{x_1x_1}G_{x_0}^{-\frac{2}{n-2}}=2-\frac{60}{n-2}Ax_1^4 +O(x_1^5) , \end{aligned}$$

and

$$\begin{aligned}&(n-2)\bigg [G_{x_0}^{-\frac{2}{n-2}} \partial _{x_1x_1}G_{x_0}^{-\frac{2}{n-2}} -\frac{1}{2}\Big( \partial _{x_1} G_{x_0}^{-\frac{2}{n-2}}\Big)^2\bigg ]-\frac{1}{2(n-1)}S_{g} G_{x_0}^{-\frac{4}{n-2}}\\&\quad =-40Ax_1^6 -\frac{1}{4(n-1)}S_{g,11}(x_0)x_1^6+O(x_1^7). \end{aligned}$$

By the definition of A, we obtain

$$\begin{aligned}&(n-2)\bigg [G_{x_0}^{-\frac{2}{n-2}} \partial _{x_1x_1}G_{x_0}^{-\frac{2}{n-2}} -\frac{1}{2}\Big( \partial _{x_1} G_{x_0}^{-\frac{2}{n-2}}\Big)^2\bigg ]-\frac{1}{2(n-1)}S_{g} G_{x_0}^{-\frac{4}{n-2}}\\&\quad =-\frac{1}{12(n-1)(n-4)}\bigg [\frac{5(n-2)}{6(n-6)}|W(x_0)|^2 -(7n-8)S_{g,11}(x_0)\bigg ]x_1^6+O(x_1^7). \end{aligned}$$

For i large, instead of (4.26), we have, at the point $x_1 e_1$,

$$\begin{aligned} R^i_{11}\le ({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}\big [-Bx_1^6+Cx_1^7+o(1)\big ], \end{aligned}$$

for some positive constant B. Then, we conclude ${R}_{11}^i$ at the point $x_1e_1$ diverges to $-\infty$ as $i\rightarrow \infty$, for $x_1>0$ sufficiently small.

Case 3.3.2. We now consider the case $W(x_0)=0$. By (4.27), we have $R_{ijkl}(x_0)=0$. Using the spherical coordinates, we set

$$\begin{aligned} \psi _i=r^i\phi _i(\theta ), \end{aligned}$$

and denote by $r^ig_i(\theta )$ the i-th Taylor expansion of $S_{g}$ at $x_0$. Let $r^lg_l(\theta )$ be the first nonzero term in the Taylor expansion of $S_{g}$ at $x_0$.

Subcase 3.3.2(a). $2\le l \le n-5$. By [13] or [24], we have $\psi _{i}=0$, $i=4, \ldots ,l-1$, and

$$\begin{aligned} {\mathcal {L}}\psi _{l+2} =-\frac{n-2}{4(n-1)} r^{l+2}g_l(\theta ), \end{aligned}$$

where

$$\begin{aligned} {\mathcal {L}}=-r^2\Delta +2(n-2)r\partial _r. \end{aligned}$$

Here, $\Delta$ is the standard Laplacian on the Euclidean space, i.e.,

$$\begin{aligned} \Delta =\frac{\partial ^2}{\partial r^2}+\frac{n-1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\Delta _{S^{n-1}}. \end{aligned}$$

Then, we have

$$\begin{aligned} (l+2)(n-4-l)\int _{S^{n-1}}\psi _{l+2}=\int _{S^{n-1}}{\mathcal {L}}\psi _{l+2} =-\frac{n-2}{4(n-1)} \int _{S^{n-1}}r^{l+2}g_l(\theta ). \end{aligned}$$

Hence,

$$\begin{aligned} \int _{S^{n-1}}\phi _{l+2}=-\frac{n-2}{4(n-1)(l+2)(n-4-l)}\int _{S^{n-1}}g_l(\theta ). \end{aligned}$$

(4.28)

We also have

$$\begin{aligned} \begin{aligned} \int _{S^{n-1}}g_l(\theta )&=\frac{r^{2-n-l}}{l}\int _{0}^{r}\int _{S^{n-1}} \Delta \big (s^l g_l(\theta )\big )s^{n-1}dr d\theta \\&=\lim _{r\rightarrow 0}\frac{r^{2-n-l}}{l}\int _{B_{r}(x_0)}\Delta _{g}S_{g}dV_{g}\\&=\lim _{r\rightarrow 0}\frac{-r^{2-n-l}}{6l}\int _{B_{r}(x_0)}|W|^2dV_{g}\le 0. \end{aligned} \end{aligned}$$

(4.29)

Note that the sign of $\lim _{r\rightarrow 0}r^{2-n-l}\int _{B_{r}(x_0)}|W|^2dV_{g}$ is independent of $g\in [g]$. Hence, if for some i with $2\le i \le n-5$,

$$\begin{aligned} 0<\lim _{r\rightarrow 0}r^{2-n-i}\int _{B_{r}(x_0)}|W|^2dV_{g}<\infty , \end{aligned}$$

then the i-th Taylor expansion of $S_{g}$ at $x_0$ must not be identical to zero.

Note

$$\begin{aligned} &G_{x_0}^{-\frac{2}{n-2}}=r^2-\frac{2}{n-2} r^{l+4}\phi _{l+2} +o( r^{l+4}),\\ &\partial _rG_{x_0}^{-\frac{2}{n-2}}=2r-\frac{2}{n-2}(l+4) r^{l+3}\phi _{l+2} +o( r^{l+3}),\\ &\partial _{rr}G_{x_0}^{-\frac{2}{n-2}}=2-\frac{2}{n-2}(l+4)(l+3)r^{l+2}\phi _{l+2}+o( r^{l+2}), \end{aligned}$$

and hence,

$$\begin{aligned}&(n-2)\bigg [G_{x_0}^{-\frac{2}{n-2}} \partial _{x_1x_1}G_{x_0}^{-\frac{2}{n-2}} -\frac{1}{2}\Big( \partial _{x_1} G_{x_0}^{-\frac{2}{n-2}}\Big)^2\bigg ]-\frac{1}{2(n-1)}S_{g} G_{x_0}^{-\frac{4}{n-2}}\\&\quad =\bigg [-2(l+2)(l+3)\phi _{l+2}(\theta )-\frac{1}{2(n-1)}g_{l}(\theta ) \bigg ]r^{l+4}+o(r^{l+4}). \end{aligned}$$

By [13, p. 61], along a radial geodesic $(\cdot ,\theta )$, $\partial ^i_rR_{rr}(x_0)=0$, $i=1, \ldots ,l-1$, and $\partial ^l_rR_{rr}(x_0)\le 0$. Therefore, for i large, by (2.9), we have, along a radial geodesic $(\cdot ,\theta )$,

$$\begin{aligned} \begin{aligned} R^i_{rr}|_{(r,\theta )}&\le ({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}\bigg \{ \bigg [-2(l+2)(l+3)\phi _{l+2} (\theta ) -\frac{1}{2(n-1)}g_{l}(\theta ) \bigg ]r^{l+4}\\&\quad\ +o(r^{l+4})+o(1)\bigg \}, \end{aligned} \end{aligned}$$

(4.30)

where o(1) denotes terms converging to zero as $i\rightarrow \infty$, uniformly for small x away from 0. By (4.28) and (4.29), we have

$$\begin{aligned}&-2(l+2)(l+3)\int _{S^{n-1}}\phi _{l+2}(\theta )d\theta -\frac{1}{2(n-1)}\int _{S^{n-1}}g_{l}(\theta )d\theta \\&\quad =\frac{(n-2)(l+3)-(n-4-l)}{2(n-1)(n-4-l)}\int _{S^{n-1}}g_l(\theta ) d\theta \le 0. \end{aligned}$$

By [13, Lemma 5.3] or [24, Chapter 5],

$$\begin{aligned} 2(l+2)(l+3)\phi _{l+2} \ne \frac{1}{2(n-1)}g_{l}. \end{aligned}$$

Hence, we can find $\theta _0 \in S^{n-1}$ such that

$$\begin{aligned} -2(l+2)(l+3)\phi _{l+2} (\theta _0)-\frac{1}{2(n-1)}g_{l}(\theta _0) \le -\epsilon _0, \end{aligned}$$

for some positive constant $\epsilon _0$. Therefore, along the radial geodesic $(r,\theta _0)$,

$$\begin{aligned} R^i_{rr}|_{(r,\theta _0)}\le ({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}\big [-\epsilon _0r^{l+4}+o(r^{l+4})+o(1)\big ]. \end{aligned}$$

Then, we conclude ${R}_{rr}^i$ at the point $(r,\theta _0)$ diverges to $-\infty$ as $i\rightarrow \infty$, for $r>0$ sufficiently small.

Subcase 3.3.2(b). $l\ge n-4$. When n is even, we have

$$\begin{aligned} {\mathcal {L}}(\psi _{n-2}+cr^{n-2} \log r)={\mathcal {L}}\psi _{n-2}-(n-2)cr^{n-2} =-\frac{n-2}{4(n-1)} r^{n-2}g_{n-4}(\theta ). \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned} (n-2)c w_{n-1}&=-r^{2-n}\int _{S^{n-1}}{\mathcal {L}}(\psi _{n-2}+cr^{n-2} \log r)\\&=\frac{(n-2)r^{6-2n}}{4(n-1)(n-4)}\int _{0}^{r}\int _{S^{n-1}}\Delta \big (s^{n-4}g_l(\theta )\big )s^{n-1}dr d\theta \\&=\lim _{r\rightarrow 0}\frac{(n-2)r^{6-2n}}{4(n-1)(n-4)}\int _{B_{r}(x_0)}\Delta _{g}S_{g}dV_{g}\\&=\lim _{r\rightarrow 0}\frac{-(n-2)r^{6-2n}}{24(n-1)(n-4)}\int _{B_{r}(x_0)}|W|^2dV_{g}\le 0. \end{aligned} \end{aligned}$$

(4.31)

We note

$$\begin{aligned} \lim _{r\rightarrow 0}r^{6-2n}\int _{B_{r}(x_0)}|W|^2dV_{g}=0, \end{aligned}$$

when n is odd, since $\int _{S^{n-1}}g_{n-2}(\theta ) d\theta =0$ when n is odd.

If $c>0$, we can proceed as in the proof of Case 3.2.1, $n=6$ and $|W(x_0)|\ne 0$, and conclude that $R^i_{rr}$ at x diverges to $-\infty$ as $i\rightarrow \infty$, for some x sufficiently close to $x_0$.

In general, we first consider the case that there exist a pair $(i,j)\in \{1,\ldots, n\}\times \{1,\ldots, n\}$ and a constant $k<[\frac{n-4}{2}]$ such that $R_{ij}\ne 0$ and k is the order of the first nonzero term in the Taylor expansion of $R_{ij}$ at $x_0$.

Without loss of generality, we assume the order of the first nonzero term in the Taylor expansion of some $R_{pq}$ at $x_0$ is k, $k<[\frac{n-4}{2}]$, and all other $R_{ij}$ vanish up to order k at $x_0$. Then, by (4.27), all $R_{ijkl}$ vanish up to order k, and hence, all $g_{ij}-\delta _{ij}$ vanish up to order $k+2$. By a rotation, we can assume

$$\begin{aligned} \frac{\partial ^k}{\partial x_1^k}R_{pq}|_{x_0}\ne 0. \end{aligned}$$

By [13, p. 61], $(p,q)\ne (1,1)$. If $p \ne 1$ and $q \ne 1$, by a rotation, we can assume $p=q=2$. Otherwise, we can assume $(p,q)=(1,2)$.

We consider the case $(p,q)=(2,2)$. By the Gauss Lemma, we have

$$\begin{aligned} x_j=\sum _{i=1}^n g_{ji}x_i. \end{aligned}$$

Then, we have, on the $x_1$-axis near $x_0=0$,

$$\begin{aligned} 1&=x_1\frac{\partial }{\partial x_2 } g_{12}+g_{22}, \end{aligned}$$

(4.32)

$$\begin{aligned} 0&=x_1\frac{\partial }{\partial x_2 } g_{11}+g_{12}, \end{aligned}$$

(4.33)

and

$$\begin{aligned} 0=x_1\frac{\partial ^2 }{\partial x_2^2}g_{22}+2\frac{\partial }{\partial x_2}g_{12}. \end{aligned}$$

We also have, at $x_0=0$,

$$\begin{aligned} (k+2)\frac{\partial ^{k+2} }{\partial x_1^{k+1} \partial x_2}g_{21}+\frac{\partial ^{k+2} }{\partial x_1^{k+2} }g_{22}=0, \end{aligned}$$

and

$$\begin{aligned} (k+1)\frac{\partial ^{k+2} }{\partial x_1^{k} \partial ^2 x_2}g_{11} +2\frac{\partial ^{k+2} }{ \partial x_1^{k+1} \partial x_2}g_{12}=0. \end{aligned}$$

Hence, we have, at $x_0$,

$$\begin{aligned} \frac{\partial ^{k} }{\partial x_1^{k} }R_{1212}&=\frac{1}{2} \bigg ( 2\frac{\partial ^{k+2} }{\partial x_1^{k+1} \partial x_2}g_{21}- \frac{\partial ^{k+2} }{\partial x_1^{k+2} }g_{22} - \frac{\partial ^{k+2}}{\partial x_1^{k} \partial ^2 x_2}g_{11}\bigg )\\&=-\frac{k+3}{k+1}\frac{\partial ^{k+2} }{\partial x_1^{k+2} }g_{22}(x_0). \end{aligned}$$

Therefore, by (4.27), we have

$$\begin{aligned} \frac{\partial ^{k} }{\partial x_1^{k} }|_{x_0}R_{22} =(n-2)\frac{\partial ^{k} }{\partial x_1^{k} }|_{x_0}R_{1212} =-(n-2)\frac{k+3}{k+1}\frac{\partial ^{k+2} }{\partial x_1^{k+2} }g_{22}(x_0)<0.\end{aligned}$$

(4.34)

Then, for i large, by (2.9), we have, at the point $x_1 e_1$,

$$\begin{aligned} R^i_{22}&=({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}\bigg [\frac{1}{k!}\frac{\partial ^{k}}{\partial x_1^{k} }|_{x_0}R_{22}x_1^{k+4} +(n-2)\big (-2\Gamma _{22}^1 x_1^3-2(g_{22}-1)x_1^2\big )\\&\quad\ +O(x_1^{k+5} )+o(1)\bigg ], \end{aligned}$$

where o(1) denotes terms converging to zero as $i\rightarrow \infty$, uniformly for small $x_1$ away from 0. At the point $x_1 e_1$,

$$\begin{aligned} \Gamma _{22}^1 =\frac{1}{2}\bigg (2\frac{\partial }{\partial x_2}g_{12}-\frac{\partial }{\partial x_1}g_{22}\bigg ). \end{aligned}$$

Combining with (4.32), we get, at the point $x_1 e_1$,

$$\begin{aligned} R^i_{22}&= ({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}\bigg [\frac{1}{k!}\frac{\partial ^{k}}{\partial x_1^{k} }|_{x_0}R_{22}x_1^{k+4} +(n-2)\frac{\partial }{\partial x_1}g_{22}x_1^3+O(x_1^{k+5} )+o(1)\bigg ]\\&=({\widetilde{u}}_i(p_0))^{-\frac{4}{n-2}}\bigg [-(n-2)\frac{k+2}{(k+1)!} \frac{\partial ^{k+2} }{\partial x_1^{k+2} }g_{22}(x_0)x_1^{k+4}+O(x_1^{k+5} )+o(1)\bigg ]. \end{aligned}$$

Then, we conclude ${R}_{22}^i$ at the point $x_1e_1$ diverges to $-\infty$ as $i\rightarrow \infty$, for $x_1>0$ sufficiently small.

If $(p,q)=(1,2)$, we can argue similarly to conclude that $|{R}_{12}^i|$ at the point $x_1e_1$ diverges to $\infty$ as $i\rightarrow \infty$, for $x_1>0$ sufficiently small.

We now consider the case that the order of the first nonzero term in the Taylor expansion of all $R_{pq}$ is greater or equal to $[\frac{n-4}{2}]$ at $x_0$, and

$$\begin{aligned} \lim _{r\rightarrow 0}r^{6-2n}\int _{B_{r}(x_0)}|W|^2dV_{g}=0. \end{aligned}$$

Then, by (4.27), the order of the first nonzero term in the Taylor expansion of $R_{ijkl}$ at $x_0$ is greater or equal to $[\frac{n-4}{2}]$, and hence, the order of the first nonzero term in the Taylor expansion of $g_{ij}-\delta _{ij}$ at $x_0$ is greater or equal to $[\frac{n-4}{2}]+2$. Hence, $({\widetilde{M}}, {\widetilde{g}})$ is asymptotically flat of order $[\frac{n-4}{2}]+2$. Thus, the ADM-mass of $({\widetilde{M}}, {\widetilde{g}})$ is well defined. By the positive mass theorem, we have

$$\begin{aligned} \int _{S^{n-1}}\big (\phi _{n-2}(\theta )+\alpha (0)\big )d\theta >0. \end{aligned}$$

Then, we can proceed as in the proof of Case 2.2, $n=6$ and $|W(x_0)|= 0$, and find $\theta _0\in S^{n-1}$ such that $R^i_{rr}$ at $(r,\theta _0)$ diverges to $-\infty$ as $i\rightarrow \infty$, for some r sufficiently small. $\square$

Remark 4.4

We point out that we used the positive mass theorem in the proof of Theorem 4.3 if the Yamabe invariant of (M, [g]) is between zero and that of the standard sphere and one of the following conditions holds: (1) M is locally conformally flat, (2) $3\le n\le 5$, or (3) for $n\ge 6$, the Weyl tensor W at $x_0$ satisfies

$$\begin{aligned} \nabla ^i|W|^2(x_0)= 0\quad \text {for any } i=0, \ldots , n-6. \end{aligned}$$

Remark 4.5

The blow-up phenomena in Theorem 4.3 are significantly different from those for the case that the underlying manifold is $S^n$. For example, take $\Omega =S^n\setminus B_{r}(e_n)$, where $B_{r}(e_n)$ is a small ball on $S^n$ centered at the north pole. Then, $\Omega$ is close to $S^n\setminus \{e_n\}$ and the complete conformal metric $g_\Omega$ in $\Omega$ with the constant scalar curvature $-n(n-1)$ has a constant sectional curvature $-1$! This can be verified by the stereographic projection, as the image of $S^n\setminus B_{r}(e_n)$ under the stereographic projection from the north pole is a ball in $\mathbb R^n$ centered at the origin.

We note that Theorem 1.3 follows easily from Theorem 4.3. Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2

Let $u_i$ be the solution of (2.1) and (2.2) in $\Omega _i$. Then, $g_{i}=u_i^{\frac{4}{n-2}}g.$

The proof of Theorem 4.2 can be adapted to prove Case 1, i.e., $\Gamma$ contains a submanifold of dimension j, with $1\le j\le \frac{n-2}{2}$.

Next, we consider Case 2, i.e., (M, g) is not conformally equivalent to the standard sphere $S^n$ and $\Gamma$ consists of finitely many points. If $\lambda (M,[g])\le 0$, the proofs of Theorem 4.2 and Theorem 4.3 can be adapted to yield the desired conclusion. Hence, we only need to discuss the case $\lambda (M,[g])> 0$ and $\Gamma$ consists of finitely many points $\{p_1, \ldots , p_k\}$.

Let $G_{p_j}\in C^{\infty }(M \setminus \{p_j\})$ be the Green’s function for the conformal Laplacian $L_g$ with the pole at $p_j$, $j=1, \ldots , k$, respectively; namely,

$$\begin{aligned} L_gG_{p_j}=(n-2)\omega _{n-1}\delta _{p_j},\quad G_{p_j}>0, \end{aligned}$$

where $\omega _{n-1}$ is the volume of $S^{n-1}$. Up to conformal factors, we assume (M, g) has conformal normal coordinates in small neighborhoods of $p_i$. Consider the metric

$$\begin{aligned} {\widetilde{g}} = \big (G_{p_1}+\cdots +G_{p_k}\big )^{\frac{4}{n-2}}g\quad \text {on }{\widetilde{M}} =M \setminus \{p_1, \ldots , p_k\}. \end{aligned}$$

Then, $({\widetilde{M}}, {\widetilde{g}})$ is an asymptotically flat and scalar flat manifold, and ${\widetilde{g}}$ has an asymptotic expansion near infinity.

Set $u_i= (G_{p_1}+\cdots +G_{p_k}){\widetilde{u}}_i$. Then, ${\widetilde{u}}_i$ satisfies

$$\begin{aligned} &\Delta _{{\widetilde{g}}} {\widetilde{u}}_{i}= \frac{1}{4}n(n-2) {\widetilde{u}}_{i}^{\frac{n+2}{n-2}} \quad \text {in }\,\Omega _i,\\ &{\widetilde{u}}_{i}=\infty \quad \text {on }\partial \Omega _i, \end{aligned}$$

and, for any m,

$$\begin{aligned} {\widetilde{u}}_i\rightarrow 0\quad \text {in }C^{m}_{\mathrm {loc}}( M \setminus \{p_1, \ldots , p_k\}) \text { as }i\rightarrow \infty .\end{aligned}$$

Fix a point $p_0\in M \setminus \{p_1, \ldots , p_k\}$. Then, for i sufficiently large, $p_0 \in \Omega _i$. Set ${\widetilde{w}}_i={{\widetilde{u}}_i}/{{\widetilde{u}}_i(p_0)}$. Then, ${\widetilde{w}}_i(p_0)=1$ and ${\widetilde{w}}_i$ satisfies

$$\begin{aligned} &\Delta _{{\widetilde{g}}} {\widetilde{w}}_{i}= \frac{1}{4}n(n-2)u_i(p_0)^{\frac{4}{n-2}} {\widetilde{w}}_{i}^{\frac{n+2}{n-2}} \quad \text {in }\,\Omega _i,\\& {\widetilde{w}}_{i}=\infty \quad \text {on }\partial \Omega _i. \end{aligned}$$

By interior estimates, there exists a positive function ${\widetilde{w}}\in {\widetilde{M}}$ such that, for any m,

$$\begin{aligned} {\widetilde{w}}_i\rightarrow {\widetilde{w}}\quad \text {in }C^{m}_{\mathrm {loc}}( {\widetilde{M}}) \text { as }i\rightarrow \infty , \end{aligned}$$

and

$$\begin{aligned}\Delta _{{\widetilde{g}}} {\widetilde{w}}=0 \quad \text {in } {\widetilde{M}}.\end{aligned}$$

Hence,

$$\begin{aligned} L_g\big ( (G_{p_1}+\cdots +G_{p_k}){\widetilde{w}}\big )=0 \quad \text {in } M \setminus \{p_1, \ldots , p_k\}. \end{aligned}$$

By the expansions of $G_{p_j}$ near $p_j$, $j=1, \ldots , k$, respectively, and [14, Proposition 9.1], we conclude that ${\widetilde{w}}$ converges to some constant $\alpha _j$ as $x\rightarrow p_j$. Without loss of generality, we assume

$$\begin{aligned} \alpha _1 \ge \alpha _2\ge \cdots \ge \alpha _k\ge 0. \end{aligned}$$

Then, $\alpha _1\ge 1$. By [14, Proposition 9.1], $(G_{p_1}+\cdots +G_{p_k}){\widetilde{w}}$ can be extended to a $C^2$-function in a neighborhood of $p_j$ if $\alpha _j=0$.

If some of $\alpha _1$, $\ldots$, $\alpha _k$ is zero, we denote by l the first integer in $\{1, \ldots , k\}$ such that $\alpha _l=0$. Otherwise, we set $l=k+1$. We always have $l\ge 2$.

Consider the metric

$$\begin{aligned} {\widehat{g}} = (G_{p_1}+\cdots +G_{p_{l-1}})^{\frac{4}{n-2}}g\quad \text {on }{\widehat{M}} =M \setminus \{p_1, \ldots , p_{l-1}\}. \end{aligned}$$

Set $(G_{p_1}+\cdots +G_{p_k}){\widetilde{w}}= (G_{p_1}+\cdots +G_{p_{l-1}}){\widehat{w}}$. Then, ${\widehat{w}}$ satisfies

$$\begin{aligned} L_g\big ((G_{p_1}+\cdots +G_{p_{l-1}}){\widehat{w}}\big )=0 \quad \text {in } M \setminus \{p_1,\cdots ,p_{l-1}\}, \end{aligned}$$

and

$$\begin{aligned} \Delta _{{\widehat{g}}} {\widehat{w}}=0 \quad \text {in } {\widehat{M}}. \end{aligned}$$

We also have that ${\widehat{w}}$ converges to $\alpha _j$ as $x\rightarrow p_j$, $j=1, \ldots , l-1$. By [14, Proposition 9.1] and the maximum principle, we have, near the point $p_{l-1}$,

$$\begin{aligned} {\widehat{w}}=\alpha _{l-1}+C_{l-1}r^{n-2}+O(r^{n-1}), \end{aligned}$$

for some nonnegative constant $C_{l-1}$. Then, the proof follows similarly as that of Theorem 4.3.

Next, we consider Case 3, i.e., (M, g) is conformally equivalent to the standard sphere $S^n$ and $\Gamma$ consists of at least two but only finitely many points. We can assume $(M,g)=(S^n,g_{S^n})$. By Lemma 4.1, we have, for any m,

$$\begin{aligned} u_i\rightarrow 0\quad \text {in }C^{m}_{\mathrm {loc}}(S^n \setminus \Gamma ) \text { as }i\rightarrow \infty . \end{aligned}$$

Set $v_i=u_i^{-\frac{2}{n-2}}$. Then,

$$\begin{aligned} v_i \text { diverges to } \infty \text { locally uniformly in } S^n \setminus\Gamma \text { as } i\rightarrow \infty . \end{aligned}$$

Take two different points $p ,q \in \Gamma$ and let $\sigma _{pq}$ be the shorter geodesic connecting p and q. Up to a conformal transform if necessary, we assume $|\sigma _{pq}|=2\epsilon$, which is less than $\frac{1}{100n}$, and $\sigma _{pq}\cap \Gamma =\{p,q\}$. We parametrize $\sigma _{pq}$ by its arc length $t\in [0, 2\epsilon ]$, with p corresponding to $t=0$ and q to $t=2\epsilon$.

For i large, let $p_i$ and $q_i$ be the points parametrized by ${\widehat{t}}_i$ and ${\widetilde{t}}_i$, respectively, where

$$\begin{aligned} {\widehat{t}}_i&=\min \{t'\in [0, \epsilon ] \,|\,te_n \in \Omega _i, \text { for any } t\in ( t',\epsilon ]\},\\ {\widetilde{t}}_i&=\max \{t'\in [\epsilon , 2\epsilon ] \,|\,t' \in \Omega _i ,\text { for any } t\in [\epsilon ,t')\}. \end{aligned}$$

Then, $p_i, q_i\in \partial \Omega _i$ and

$$\begin{aligned} p_i\rightarrow p,\quad q_i\rightarrow q. \end{aligned}$$

For convenience, we denote by $v_i(t)$ the function $v_i$ restricted to the geodesic $\sigma _{pq}$. By the polyhomogenous expansion of $v_i$, we have $|\partial _t v_i(p_i)|\le 1$ and $|\partial _t v_i(q_i)|\le 1$.

Since $v_i(\epsilon )\rightarrow \infty$, for i large, we take $t_i \in ({\widehat{t}}_i,{\widetilde{t}}_i)$ such that, for any $t\in ({\widehat{t}}_i,{\widetilde{t}}_i)$,

$$\begin{aligned} \partial _tv_i(t)\le \partial _tv_i(t_i). \end{aligned}$$

Then,

$$\begin{aligned} \partial _tv_i(t_i)>\frac{v_i(\epsilon )-0}{\epsilon -{\widehat{t}}_i}\ge \frac{ v_i(\epsilon )}{\epsilon }, \end{aligned}$$

and

$$\begin{aligned} \partial _{tt}v_i( t_i)=0. \end{aligned}$$

Denote by $R^i_{tt}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\frac{\partial }{\partial t}$ with respect to the metric $g_i$. Then, we can verify at the point $t_ie$, $R^i_{tt}$ diverges to $-\infty$ as $i\rightarrow \infty$. $\square$

5 General Domains in Euclidean Spaces

In this section, we present several examples of smooth bounded domains $\Omega$ in the Euclidean space and examine whether the complete conformal metrics $g_\Omega$ associated with the Loewner–Nirenberg problem have negative Ricci curvatures. We demonstrate by these examples the complexity of the issue studied in this paper. Topological conditions are not sufficient to determine whether these complete conformal metrics have negative Ricci curvatures.

There are two classes of examples. First, we construct nonconvex smooth domains in which the complete conformal metrics with a constant scalar curvature still have negative Ricci curvatures. Second, we construct bounded smooth domains in which the complete conformal metrics have positive Ricci components at some points.

By the Cartan–Hadamard Theorem, we know that $\pi _{i}(\Omega )=0$ for $i\ge 2$ if $g_\Omega$ has negative sectional curvatures in $\Omega$, where $\pi _{i}(\Omega )$ is the i-th homotopy group of $\Omega$. The following example shows that $\pi _{1}(\Omega )=0$ is not necessary for $g_\Omega$ to have negative sectional curvatures.

Example 5.1

Set

$$\begin{aligned} \Omega _r=\Bigg \{(x_1, \ldots , x_n)\,\bigg|\,\bigg (x_1-\frac{x_1}{\sqrt{x_1^2+x_n^2}}\bigg )^2+\sum _{i=2}^{n-1}x_i^2 + \bigg (x_n-\frac{x_n}{\sqrt{x_1^2+x_n^2}}\bigg )^2 < r^2 \Bigg \}\subset {\mathbb {R}}^n, \end{aligned}$$

where $0<r<{1}/{100}$. Then, $\pi _{1}(\Omega _r)= {\mathbb {Z}}$ and $\pi _{i}(\Omega )=0$ for $i\ge 2$. We claim that $g_{\Omega _r}$ has negative sectional curvatures in $\Omega _r$, if r is sufficiently small. By symmetry, it suffices to show $g_{\Omega _r}$ has negative sectional curvatures in

$$\begin{aligned} \Bigg \{(x_1, \ldots , x_n)\,|\,(x_1-1)^2+\sum _{i=2}^{n-1}x_i^2<r^2, x_n=0 \Bigg \}\subset {\mathbb {R}}^n. \end{aligned}$$

Note that $\Omega _r$ is transformed to

$$\begin{aligned}{\widetilde{\Omega }}_r&=\Bigg \{y\in {\mathbb {R}}^n\,\bigg|\,\bigg (y_1+\frac{1}{r}-\frac{ry_1+1}{r\sqrt{(ry_1+1)^2+(ry_n)^2}}\bigg )^2\\&\qquad +\sum _{i=2}^{n-1}y_i^2+ \bigg (y_n-\frac{y_n}{\sqrt{(ry_1+1)^2+(ry_n)^2}}\bigg )^2 < 1 \Bigg \}, \end{aligned}$$

under the transform

$$\begin{aligned} y_1=\frac{x_1-1}{r},\, y_2=\frac{x_2}{r},\,\ldots ,\,y_n=\frac{x_n}{r}. \end{aligned}$$

Then, $\{ {\widetilde{\Omega }}_r\}$ converges in $C^{k}$, for any integer $k\ge 1$, to

$$\begin{aligned} \Omega _0=\{(x_1, \ldots , x_n)\,|\,x_1^2+\cdots +x_{n-1}^2<1 \}\subset {\mathbb {R}}^n \end{aligned}$$

in any compact sets in ${\mathbb {R}}^n$. Let $u^0$ be the positive solution of (2.11)−(2.12) for $\Omega =\Omega _0$. Then, $u^0$ has the form

$$\begin{aligned} u^0(x_1, \ldots ,x^n)=u^0(x_1, \ldots ,x_{n-1},0). \end{aligned}$$

By the same method as in the proof of Theorem 1.1, we can prove $(u^0)^{\frac{4}{n-2}}|dx|^2$ has negative sectional curvatures in $\Omega _0$. Then, the polyhomogeneous expansions for $u_r$ imply that $g_{{\widetilde{\Omega }}_r}$ has negative sectional curvatures in

$$\begin{aligned} \{(x_1, \ldots , x_n)\,|\,1-\delta<x_1^2+\cdots +x_{n-1}^2<1, x_n=0 \}, \end{aligned}$$

for some small $\delta >0$, independent of r. It is straightforward to prove that $g_{{\widetilde{\Omega }}_r}$ has negative sectional curvatures in

$$\begin{aligned} \{(x_1, \ldots , x_n)\,|\,x_1^2+\cdots +x_{n-1}^2\le 1-\delta , x_n=0 \}, \end{aligned}$$

since $\{{\widetilde{\Omega }}_r \}$ converges to $\Omega _0$ in any compact sets in ${\mathbb {R}}^n$.

In the next example, we construct a bounded smooth domain $\Omega \subset {\mathbb {R}}^n$ which is diffeomorphic to the unit ball and cannot be conformally transformed to a bounded convex domain such that the complete conformal metric $g_\Omega$ with a negative scalar curvature possesses negative sectional curvatures in $\Omega$.

Example 5.2

Let $\sigma$ be a U-shaped smooth curve in ${\mathbb {R}}^n$ with two endpoints p and q. Let $\Omega ^{r}$, $0<r<{1}/{100}$, be a family of tubular domains with smooth boundaries satisfying the following conditions:

• (A1) $\Omega ^{r_2}\subseteq \Omega ^{r_1}$ if $r_2 <r_1$;

• (A2) $\bigcap \Omega ^{r} =\sigma$;

• (A3) For a fixed point $x_0 \in \sigma \setminus \{p,q\}$, $\{x\in {\mathbb {R}}^n\,|\,rx+x_0\in \Omega ^r\}$ converges in $C^{k}$, for any integer $k\ge 1$, to a smooth domain which is equal to $\{(x_1, \ldots , x_n)\,|\,x_1^2+\cdots +x_{n-1}^2<1 \}\subset {\mathbb {R}}^n$ up to a Euclidean transformation in any compact set in ${\mathbb {R}}^n$ as $r\rightarrow 0$;

• (A4) At the point p or q, $\{x\in {\mathbb {R}}^n\,|\,rx+x_0\in \Omega ^r\}$ converges in $C^{k}$, for any integer $k\ge 1$, to a smooth domain which is equal to $\Omega ^0$ up to a Euclidean transformation in any compact set in ${\mathbb {R}}^n$ as $r\rightarrow 0$, where $\Omega ^0$ is a smooth convex domain which coincides $\{(x_1, \ldots , x_n)\,|\,x_1^2+\cdots +x_{n-1}^2<1 \}$ when $x_n\ge 0$, and coincides $\{(x_1, \ldots , x_n)\,|-\sqrt{x_1^2+\cdots +x_{n-1}^2}<x_n<0\}$ when $-1<x_n<{1}/{3}$.

Then, $\Omega ^{r}$ cannot be conformally transformed to bounded convex domains if r is sufficiently small. Otherwise, there would exist an arc $\sigma _r$ with endpoints p and q such that $\sigma _r \subseteq \Omega _r$. Let w be the positive solution of (2.11)–(2.12) for $\Omega =\Omega ^0$. Then by approximation and using the same method as in the proof of Theorem 1.1, we can prove $w^{-\frac{2}{n-2}}$ is concave in $\Omega ^0$ and $w^{\frac{4}{n-2}}|dx|^2$ has negative sectional curvatures in $\Omega ^0$. Arguing as in Example 5.1, we can show that the sectional curvatures in $\Omega ^0$ is close to the sectional curvatures in $\Omega _0$ when $x_n$ is sufficiently large. Hence, the sectional curvatures in $\Omega ^0$ are bounded above by a negative constant. Then, arguing as in Example 5.1 again, we obtain that $g_{\Omega ^r}$ has negative sectional curvatures in $\Omega ^r$ when r is sufficiently small.

Example 5.3

For $R>r>0$, consider the annular region $\Omega _{R,r}=B_{R}\setminus B_r$ in ${\mathbb {R}}^n$. Let $g_{\Omega _{R,r}}$ be the complete conformal metric with the constant scalar curvature $-n(n-1)$ in $\Omega _{R,r}$. Arguing as in Example 5.1, we can prove that the maximum Ricci curvature component of $g_{\Omega _{R,r}}$ is close to $-{n}/{2}$ as R and r tend to 1. Hence, $g_{\Omega _{R,r}}$ has negative Ricci curvatures in $\Omega _{R,r}$ as R is sufficiently close to r.

In the rest of this section, we construct bounded domains in which the complete conformal metrics have positive Ricci components at some points. The most straightforward way to do this is to combine Theorem 1.2 for the case $(M,g)=(S^n,g_{S^n})$ and the stereographic projections.

We identify ${\mathbb {R}}^{n}$ in ${\mathbb {R}}^{n+1}$ as ${\mathbb {R}}^{n}\times \{0\}$ and write $x=(x_1, \ldots , x_n)\in {\mathbb {R}}^n$. Then,

$$\begin{aligned} S^n=\{(x,x_{n+1}):\,|x|^{2}+x_{n+1}^{2}=1\}. \end{aligned}$$

Consider the transform $T: {{\mathbb {R}}^{n} \rightarrow {S}^{n}}$ given by

$$\begin{aligned} T(x)=\bigg (\frac{2 x}{1+|x|^2},\frac{|x|^2-1}{1+|x|^2}\bigg ). \end{aligned}$$

Then, T is the inverse transform of the stereographic projection which lifts ${\mathbb {R}}^{n}\times \{0\}$ to $S^n$.

Proposition 5.4

Let $\Gamma$ be a set in $S^n$ as in Theorem 1.2, containing the north pole. Suppose ${\widetilde{\Omega }}_i$ is a sequence of increasing smooth domains in $S^n$ which converges to $S^{n}\setminus \Gamma$, with $\partial {\widetilde{\Omega }}_i$ not containing the north pole, and set $\Omega _i=T^{-1}({\widetilde{\Omega }}_i)$. Assume $g_i$ is the complete conformal metric in $\Omega _i$ with the constant scalar curvature $-n(n-1)$. Then, for sufficiently large i, $g_i$ has a positive Ricci curvature component somewhere in $\Omega _{i}$. Moreover, the maximal Ricci curvature of $g_i$ in $\Omega _{i}$ diverges to $\infty$ as $i\rightarrow \infty$.

Proposition 5.4 follows easily from Theorem 1.2 for the case $(M,g)=(S^n,g_{S^n})$.

We point out that notations in Proposition 5.4 is slightly different from those in Theorem 1.2. In Proposition 5.4, ${\widetilde{\Omega }}_i$ is a domain in $S_n$ and $\Omega _i$ is a domain in ${\mathbb {R}}^n$. We also note that $\Omega _i$ is a bounded domain in ${\mathbb {R}}^n$ if the north pole is not in the closure of ${\widetilde{\Omega }}_i$.

Example 5.5

Let $\{p_1,\ldots ,p_k \}$ be a collection of finitely many points in ${\mathbb {R}}^n$, with $k\ge 1$, and set $\Omega _{R,r}=B_R(0)\backslash \bigcup _{i=1}^{k} B_r(p_i)$. Then, for R sufficiently large and r sufficiently small, the complete conformal metric in $\Omega _{R,r}$ with the constant scalar curvature $-n(n-1)$ has a positive Ricci curvature component somewhere. Note that the corresponding $\Gamma$ in $S^n$ is given by $\Gamma =\{e_{n+1}, T(p_1), \ldots , T(p_k)\}$, which consists of at least two points. If $k=1$ and $p_1=0$, then $\Omega _{R, r}$ is the annular region as in Example 5.3. Combining with Example 5.3 for a fixed constant r, we conclude that the maximum Ricci curvature component of $g_{\Omega _{R,r}}$ tends to $-{n}/{2}$ as $R\rightarrow r$ and tends to $\infty$ as $R\rightarrow \infty$.

Next, we construct bounded star-shaped domains in which the complete conformal metrics have positive Ricci components somewhere.

Example 5.6

For $n\ge 4$, set

$$\begin{aligned} \gamma =\{(0, \ldots , 0, x_n)\,|\, |x_n|\ge 1\}\subset {\mathbb {R}}^n. \end{aligned}$$

Let $\Omega _i$ be a sequence of increasing bounded smooth domains in ${\mathbb {R}}^n$, star-shaped with respect to the origin, which converges to ${\mathbb {R}}^n\setminus \gamma$. Then, for i sufficiently large, the complete conformal metric in $\Omega _i$ with the constant scalar curvature $-n(n-1)$ has a positive Ricci curvature component somewhere. Note that the corresponding $\Gamma$ in $S^n$ is given by the equator in the $x_n$-$x_{n+1}$ plane minus the image under T of the segment $(-1,1)$ on $x_n$-axis. Hence, the dimension of $\Gamma$ is 1. This is the reason we require $n\ge 4$. We point out that domains in this example are diffeomorphic to balls.