Energy, Hopf Differential, and Metrics on Teichmüller Space

Abstract

Let $T(X)$ be the Teichmüller space of a closed surface $X$ with genus $g \ge 2$. For a given point $[\sigma ] \in T(X)$, let $Q(\sigma )$ be the space of holomorphic quadratic differentials on $(X, \sigma )$. Denote by $\Phi : T(X) \rightarrow Q(\sigma )$ the Hopf mapping which maps a point $[\rho ] \in T(X)$ to the Hopf differential of the unique harmonic map $w: (X, \sigma ) \rightarrow (X, \rho )$ in the homotopy class of the identity $id: (X, \sigma ) \rightarrow (X, \rho )$. In this paper, we give quantitative comparisons of both the energy and $L_1$-norm of the Hopf differential involved in $\Phi$ with the length spectrum metric, the Teichmüller metric, and Thurston’s asymmetric metrics, respectively.

1 Introduction

Let us introduce some notations first, for details see Sect. 2 and the references therein. Let $X$ be a closed surface of genus $g \ge 2$ and $T(X)$ the Teichmüller space of $X$. For a given $[\sigma ] \in T (X)$, let $Q(\sigma )$ be the space of holomorphic quadratic differentials on $(X, \sigma )$, endowed with the $L_1$-norm $|| \cdot ||_1$. In this paper, we are concerned with the Hopf mapping (cf. [15]) $\Phi : T(X) \rightarrow Q(\sigma )$ which maps a point $[\rho ] \in T(X)$ to the Hopf differential of the unique harmonic map $w: (X, \sigma ) \rightarrow (X, \rho )$ in the homotopy class of the identity $id: (X, \sigma ) \rightarrow (X, \rho )$. It is now well known that [15] $\Phi : (T(X), d_\mathrm{{T}}) \rightarrow (Q(\sigma ), || \cdot ||_1)$ is a homeomorphism, where $d_\mathrm{{T}}$ is the Teichmüller metric on $T (X)$.

There are many interesting metrics on $T (X)$, among them we have (cf. [1, 8, 9, 12–14]) the length spectrum metric $d_\mathrm{{L}}$, the Teichmüller metric $d_\mathrm{{T}}$, and Thurston’s asymmetric metrics $d_{\mathrm{{P}}_i}$, $i = 1, 2$. In this paper, we will give quantitative comparisons of both the energy $E(\sigma , \rho )$ of the harmonic map $w$ and the $L_1$-norm of the Hopf differential $\Phi (\rho )$ with each of the above metrics. The main results are as follows:

Theorem 1

There exists a constant $C_1$ depending only on the Euler characterization $\chi (X)$ of $X$ such that

$$\begin{aligned} E (\sigma , \rho )&\le \pi |\chi (X)| e^{2 d_{\mathrm{{P}}_1} (\sigma , \rho )} + C_1 \\&\le \pi |\chi (X)| e^{2 d_\mathrm{{L}} (\sigma , \rho )} +C_1 \\&\le \pi |\chi (X)| e^{2 d_\mathrm{{T}} (\sigma , \rho )} +C_1 \end{aligned}$$

holds for any two points $[\sigma ]$ and $[\rho ]$ in $T(X)$.

For any $[\sigma ]$ and $[\rho ]$ in $T(X)$,

$$\begin{aligned} E (\sigma , \rho ) \le C_2 (\sigma ) \pi |\chi (X)| e^{4 d_{\mathrm{{P}}_2} (\sigma , \rho )} +C_1, \end{aligned}$$

where $C_2 (\sigma )$ is a constant depending only on $[\sigma ]$.

Theorem 2

There exists a constant $C_3$ depending only on $\chi (X)$ such that

$$\begin{aligned} ||\Phi (\rho )||_1&\le \frac{\pi |\chi (X)|}{2} e^{2 d_{\mathrm{{P}}_1} (\sigma , \rho )} + C_3 \\&\le \frac{\pi |\chi (X)|}{2} e^{2 d_\mathrm{{L}} (\sigma , \rho )} +C_3 \\&\le \frac{\pi |\chi (X)|}{2} e^{2 d_\mathrm{{T}} (\sigma , \rho )} +C_3 \end{aligned}$$

holds for any two points $[\sigma ]$ and $[\rho ]$ in $T(X)$.

For any $[\sigma ]$ and $[\rho ]$ in $T(X)$,

$$\begin{aligned} ||\Phi (\rho )||_1 \le C_4 (\sigma ) \pi |\chi (X)| e^{4 d_{\mathrm{{P}}_2} (\sigma , \rho )} +C_5, \end{aligned}$$

where $C_4 (\sigma )$ is a constant depending only on $[\sigma ]$ and $C_5$ is a constant depending only on $\chi (X)$.

2 Preliminaries

2.1 Metrics on Teichmüller Space

Let $X$ be a closed surface of genus $g \ge 2$. The Teichmüller space $T(X)$ is (cf. [1, 6]) the space of equivalence classes $[S,f]$ of marked Riemann surfaces, where two marked Riemann surfaces $(S_1, f_1: X \rightarrow S_1)$ and $(S_2, f_2: X \rightarrow S_2)$ are equivalent if there exists a conformal mapping $c: S_1 \rightarrow S_2$ which is homotopic to $f_2 \circ f_1^{-1}$. By the uniformization theorem, $T(X)$ can also be viewed as the space of isotopy classes of hyperbolic metrics on $X$, where two hyperbolic metrics are isotopic if there exists an isometry between them which is isotopic to the identity. In the sequel, we will sometimes denote a point $[S, f]$ in $T(X)$ simply by $S$ (or equivalently by the hyperbolic metric $\rho$ corresponding to $S$).

The Teichmüller metric is defined as [1]

$$\begin{aligned} d_\mathrm{{T}}([S_1,f_1], [S_2,f_2])=\log \{ \inf K(f) \}, \end{aligned}$$

where the infimum is taken over all $f: S_1\rightarrow S_2$ in the homotopy class of $f_2 \circ f_1^{-1}$, and $K(f)$ is the maximal dilatation of $f$.

For (a homotopy class of) an essential closed curve $\gamma$ and $S \in T(X)$, let $l_S (\gamma )$ be the hyperbolic length of $\gamma$. Let $\Sigma _S$ be the set of homotopy classes of essential closed curves on $S$. Then the length spectrum metric $d_\mathrm{{L}}$ is defined as [12, 13]

$$\begin{aligned} d_\mathrm{{L}} ([S_1, f_1], [S_2, f_2])=\log \sup _{\gamma \in \Sigma _{S_1}} \left\{ \frac{l_{S_2}(f(\gamma ))}{l_{S_1}(\gamma )}, \frac{l_{S_1}(\gamma )}{l_{S_2}(f(\gamma ))}\right\} , \end{aligned}$$

where $f=f_2\circ f_1^{-1}$.

Thurston’s asymmetric metrics $d_{\mathrm{{P}}_1}$ and $d_{\mathrm{{P}}_2}$ are defined as [14]

$$\begin{aligned} d_{\mathrm{{P}}_1}([S_1,f_1],\ [S_2,f_2])= \log \sup _{\gamma \in \Sigma _{S_1}} \left\{ \frac{l_{S_2}(f(\gamma ))}{l_{S_1}(\gamma )}\right\} \end{aligned}$$

and

$$\begin{aligned} d_{\mathrm{{P}}_2}([S_1,f_1],\ [S_2,f_2])= \log \sup _{\gamma \in \Sigma _{S_1}} \left\{ \frac{l_{S_1}(\gamma )}{l_{S_2}(f(\gamma ))}\right\} , \end{aligned}$$

where $f=f_2\circ f_1^{-1}$. Both $d_{\mathrm{{P}}_1}$ and $d_{\mathrm{{P}}_2}$ satisfy [14] the positive definiteness and triangle inequality in the definition of a metric. However, they are not symmetric [14], namely

$$\begin{aligned} d_{\mathrm{{P}}_i} ([S_1,f_1], [S_2,f_2])= d_{\mathrm{{P}}_i}([S_2,f_2],[S_1,f_1]), \ i=1, 2 \end{aligned}$$

do not hold generally.

The following well-known result of Wolpert [16] describes length distortions under quasiconformal mappings.

Lemma 1

([16], Lemma 3.1) Let $f: S_1 \rightarrow S_2$ be a $K$-quasiconformal mapping between hyperbolic Riemann surfaces, then

$$\begin{aligned} \frac{l_{S_2}(f(\gamma ))}{l_{S_1}(\gamma )} \le K(f) \end{aligned}$$

holds for all closed curves $\gamma \subset S_1$.

By this lemma, we have the following comparisons of the above metrics.

Lemma 2

$$\begin{aligned} d_{\mathrm{{P}}_i} \le d_\mathrm{{L}} \le d_\mathrm{{T}}, \ i=1, 2. \end{aligned}$$

2.2 Energy, Harmonic Map, and Hopf Mapping

We recall briefly some relevant definitions and results (cf. [15]). Consider two metrics $\sigma (z) |\mathrm{{d}}z|^2$ and $\rho (w) |\mathrm{{d}}w|^2$ on $X$. For a Lipschitz map

$$\begin{aligned} w: \big (X, \sigma (z) |\mathrm{{d}}z|^2\big ) \rightarrow \big (X, \rho (w) |\mathrm{{d}}w|^2\big ), \end{aligned}$$

the energy of $w$ is defined by

$$\begin{aligned} E(w; \sigma , \rho )= \int _X \big (\rho (w(z)) |w_z|^2 + \rho (w(z)) |w_{\bar{z}}|^2\big ) \mathrm{{d}}x\mathrm{{d}}y. \end{aligned}$$

A critical point of the energy functional is called a harmonic map. It is characterized by the Euler–Lagrange equation

$$\begin{aligned} w_{z{\bar{z}}} +{(\log \rho )}_{w} w_z w_{\bar{z}} =0. \end{aligned}$$

For two points $[\sigma ]$ and $[\rho ]$ in $T(X)$, it is known [2, 5] that there is a unique harmonic map $w$ which is homotopic to the identity map

$$\begin{aligned}id: (X, \sigma (z) |\mathrm{{d}}z|^2) \rightarrow (X, \rho (w) |\mathrm{{d}}w|^2). \end{aligned}$$

Let $E(\sigma , \rho )$ be the energy of such a harmonic map. Note that $E(\sigma , \rho )$ depends only on the conformal class of the domain metric while on the isometry class of the target metric. In what follows, we will sometimes use the notation $E(\rho )$ to denote the energy $E (\sigma , \rho )$, in the case that $\sigma \in T(X)$ is a fixed base point with the identification $T(X)=T(\sigma )$.

For $[\sigma ] \in T (X)$, let $Q (\sigma )$ be the space of holomorphic quadratic differentials on $(X, \sigma )$, endowed with the $L_1$-norm $||\cdot ||_1$. Naturally associated to a harmonic map $w: (X, \sigma (z) |\mathrm{{d}}z|^2) \rightarrow (X, \rho (w) |\mathrm{{d}}w|^2)$ is a holomorphic quadratic differential $\phi (\sigma , \rho )$, the so-called Hopf differential, which is given by

$$\begin{aligned} \phi (\sigma , \rho )=\rho w_{z} \overline{w_{\bar{z}}}. \end{aligned}$$

Let $[\sigma ] \in T(X)$ be fixed, then we have [15] a mapping $\Phi : T(X) \rightarrow Q(\sigma )$ which maps $[\rho ] \in T(X)$ to the Hopf differential $\phi (\sigma , \rho )$ of the unique harmonic map $w$ in the homotopy class of the identity. This mapping is called the Hopf mapping. It has been known that $\Phi : (T(X), d_\mathrm{{T}}) \rightarrow (Q(\sigma ), {|| \cdot ||}_{1})$ is continuous [3, 15], injective [11, 15]. In ([15], Theorem 3.1), Wolf showed that $\Phi : (T(X), d_\mathrm{{T}}) \rightarrow (Q(\sigma ), {|| \cdot ||}_{1})$ is proper, and consequently a homeomorphism by Brouwer’s invariance of domain.

We need the following result of Minsky [10], which gives an interpretation of the energy in terms of the hyperbolic length and the extremal length.

Theorem A

([10], Theorem 7.2) There exist a constant $C$ depending only on $\chi (X)$ and a measured foliation ${\mathcal {F}}_h$, such that

$$\begin{aligned} \frac{1}{2} \frac{l_\rho ^2 ({\mathcal {F}}_h)}{ext_\sigma ({\mathcal {F}}_h)} \le E(\sigma , \rho ) \le \frac{1}{2} \frac{l_\rho ^2 ({\mathcal {F}}_h)}{ext_\sigma ({\mathcal {F}}_h)}+ C, \end{aligned}$$

where $l_\rho ({\mathcal {F}}_h)$ is the hyperbolic length of ${\mathcal {F}}_h$ and $ext_\sigma ({\mathcal {F}}_h)$ is the extremal length of ${\mathcal {F}}_h$.

Another result that we need is the following standard one due to Thurston.

Lemma 3

(cf. [4]) Positive real multiples of homotopy classes of simple closed curves are dense in the space ${\mathcal {M}} {\mathcal {F}}$ of measured foliations. Namely, for any ${\mathcal {F}} \in {\mathcal {M}} {\mathcal {F}}$, there exists a sequence of positive real numbers $r_n$ and a sequence of simple closed curves $\gamma _n$ such that $r_n \gamma _n \rightarrow {\mathcal {F}}$ $(n \rightarrow \infty )$ in ${\mathcal {M}}{\mathcal {F}}$.

3 Energy and Metrics on Teichmüller Space

In this section, we will give quantitative comparisons of the energy with the length spectrum metric, the Teichmüller metric, and Thurston’s asymmetric metrics, respectively.

Theorem 1

There exists a constant $C_1$ depending only on the Euler characterization $\chi (X)$ of $X$ such that

$$\begin{aligned} E (\sigma , \rho )&\le \pi |\chi (X)| e^{2 d_{\mathrm{{P}}_1} (\sigma , \rho )} + C_1 \nonumber \\&\le \pi |\chi (X)| e^{2 d_\mathrm{{L}} (\sigma , \rho )} +C_1 \\&\le \pi |\chi (X)| e^{2 d_\mathrm{{T}} (\sigma , \rho )} +C_1\nonumber \end{aligned}$$

(1)

holds for any two points $[\sigma ]$ and $[\rho ]$ in $T(X)$.

For any $[\sigma ]$ and $[\rho ]$ in $T(X)$,

$$\begin{aligned} E (\sigma , \rho ) \le C_2 (\sigma ) \pi |\chi (X)| e^{4 d_{\mathrm{{P}}_2} (\sigma , \rho )} +C_1, \end{aligned}$$

(2)

where $C_2 (\sigma )$ is a constant depending only on $[\sigma ]$.

Proof

Recall that [1, 10] the extremal length is defined as

$$\begin{aligned} ext_\sigma ({\mathcal {F}}_{h})= \sup _{\lambda } \frac{{(l_{\lambda } ({\mathcal {F}}_{h}))}^2}{A (X, \lambda )}, \end{aligned}$$

where the supremum is taken over all the conformal metrics $\lambda = \lambda (z) |{\text {d}}z|^2$ in the conformal equivalence class of $\sigma$ with area $0< A(X, \lambda )=\int _X \lambda < \infty$. Thus if we take the particular choice $\sigma$ in the supremum, then we get from Theorem A that

$$\begin{aligned} E(\sigma , \rho )&\le \frac{1}{2} \frac{l_\rho ^2 ({\mathcal {F}}_h)}{ext_\sigma ({\mathcal {F}}_h)}+ C \nonumber \\&\le \pi |\chi (X)| \frac{l_\rho ^2 ({\mathcal {F}}_h)}{l_\sigma ^2 ({\mathcal {F}}_h)} + C, \end{aligned}$$

(3)

where we recall from the Gauss–Bonnet formula $A (X, \sigma )= 2 \pi |\chi (X)|$.

Note that Thurston [14] proved that the suprema in the definitions of $d_{\mathrm{{P}}_i}$ $(i=1, 2)$ can also be equivalently taken over all simple closed curves. Thus, from Lemma 3, we have

$$\begin{aligned} d_{\mathrm{{P}}_1} (\sigma , \rho )= \log \left\{ \sup _{\mathcal {F}} \frac{l_\rho ({\mathcal {F}})}{l_\sigma ({\mathcal {F}})} \right\} , \end{aligned}$$

(4)

where the supremum is taken over all the measured foliations ${\mathcal {F}}$. Combining (3) and (4), we obtain

$$\begin{aligned} E (\sigma , \rho ) \le \pi |\chi (X)| e^{2 d_{\mathrm{{P}}_1} (\sigma , \rho )} +C. \end{aligned}$$

Together with Lemma 2, this inequality leads to (1).

However, since none of $E (\sigma , \rho )$ and $d_{\mathrm{{P}}_i} (\sigma , \rho )$ $(i=1, 2)$ are symmetric in their arguments, we cannot obtain (2) similarly. To show (2), we claim that

$$\begin{aligned} d_\mathrm{{T}} (\sigma , \rho ) \le 2 d_{\mathrm{{P}}_2} (\sigma , \rho ) + M (\sigma ) \end{aligned}$$

(5)

holds for any $[\sigma ]$ and $[\rho ]$ in $T(X)$, where $M (\sigma )$ is a constant depending only on $[\sigma ]$. Together with (1), this claim yields (2).

We are left to show the claim (5). To this end, we give the following general discussion. Recall that

$$\begin{aligned} d_{\mathrm{{P}}_2} (\sigma , \rho ) = \log \left\{ \sup _\gamma \frac{l_\sigma (\gamma )}{l_\rho (\gamma )} \right\} , \end{aligned}$$

while Kerckhoff [7] shows that

$$\begin{aligned} d_\mathrm{{T}} (\sigma , \rho ) = \log \left\{ \sup _{\gamma } \frac{ext_\sigma (\gamma )}{ext_\rho (\gamma )} \right\} , \end{aligned}$$

where the supremum is taken over all the simple closed curves $\gamma$. Thus in order to show the claim, we need to compare the hyperbolic length $l_\mathrm{{R}} (\gamma )$ with the extremal length $ext_\mathrm{{R}} (\gamma )$ on a Riemann surface $R$. For this, from the extremality [7] in the extremal length ${\text {ext}}_\mathrm{{R}} (\gamma )$ of the metric induced by the unit $L_1$-normed Jenkins–Strebel differential $\varphi =\varphi (z) {\text {d}}z^2$ of $\gamma$ on $R$, we get

$$\begin{aligned} \frac{l_\mathrm{{R}}^2 (\gamma )}{2 \pi |\chi (R)|} \le {\text {ext}}_\mathrm{{R}} (\gamma ) =\frac{l_{\varphi }^2 (\gamma )}{||\varphi ||_1} =l_{\varphi }^2 (\gamma ), \end{aligned}$$

where $l_{\varphi } (\gamma ) = \inf \limits _\alpha \left\{ \int _{\alpha } \sqrt{|\varphi (z)|} |{\text {d}}z| \right\}$ with the infimum taken over all the simple closed curves $\alpha$ in the homotopy class of $\gamma$. To proceed, we have the following estimate:

$$\begin{aligned} \int _{\alpha } \sqrt{|\varphi (z)|} |\mathrm{{d}}z|&= \int _{\alpha } \frac{\sqrt{|\varphi (z)|}}{\sqrt{\lambda (z)}} \sqrt{\lambda (z)} |\mathrm{{d}}z| \\&\le \sqrt{||\varphi ||_{\infty }} \int _{\alpha } \sqrt{\lambda (z)} |{\text {d}}z|, \end{aligned}$$

where $\lambda = \lambda (z) |\mathrm{{d}}z|^2$ is the hyperbolic area element on $R$ and $||\varphi ||_{\infty }=\sup \limits _\mathrm{{R}} \frac{|\varphi (z)|}{\lambda (z)}$ is Bers’ sup-norm of $\varphi$ which is finite. It follows that

$$\begin{aligned} l_{\varphi } (\gamma ) \le \sqrt{|| \varphi ||_{\infty }} \ l_\mathrm{{R}} (\gamma ). \end{aligned}$$

Furthermore, since the space $Q (R)$ of holomorphic quadratic differentials on $R$ is a finite dimensional ($3g-3$ if the genus of $R$ is $g$) Banach space, the norms $|| \cdot ||_1$ and $|| \cdot ||_{\infty }$ are equivalent, i.e., there exists a constant ${\mathcal {L}} = {\mathcal {L}} (R)$ depending only on $R$ such that

$$\begin{aligned} \frac{|| q ||_1}{\mathcal {L}} \le || q ||_{\infty } \le \mathcal {L} || q ||_1 \end{aligned}$$

holds for any $q \in Q (R)$. In particular, $|| \varphi ||_{\infty } \le \mathcal {L}$. Therefore, we conclude from the above discussion that

$$\begin{aligned} \frac{l_\mathrm{{R}}^2 (\gamma )}{2 \pi |\chi (R)|} \le {\text {ext}}_\mathrm{{R}} (\gamma ) \le {\mathcal {L}} (R) \ l_\mathrm{{R}}^2 (\gamma ). \end{aligned}$$

Consequently, claim (5) follows from this last inequality. $\square$

4 $L_1$-Norm and Metrics on Teichmüller Space

In this section, we will give quantitative comparisons of the $L_1$-norm of the Hopf differential with the aforementioned metrics. To do this, we need to compare the $L_1$-norm with the energy.

Lemma 4

$$\begin{aligned} 2 {|| \Phi (\rho ) ||}_1 \le E(\rho ). \end{aligned}$$

Proof

Let $w: \sigma (z) |{\text {d}}z|^2 \rightarrow \rho (w) |{\text {d}}w|^2$ be the unique harmonic map in the homotopy class of the identity, where $z=x+ i y$. Then from definitions we have

$$\begin{aligned} \begin{array}{ll} {\text {energy of } w:} &{} E (\rho ) = \int _X \rho (w (z)) ({|w_z|}^2 + {|w_{\bar{z}}|}^2) {\text {d}}x {\text {d}}y,\\ {\text {Hopf differential of } w:} &{} \Phi (\rho ) = \phi (\sigma , \rho ) = \rho (w (z)) w_z \overline{w_{\bar{z}}} {\text {d}}z^2. \end{array} \end{aligned}$$

Consequently

$$\begin{aligned} ||\Phi (\rho )||_1 \le \frac{1}{2} E (\rho ). \end{aligned}$$

$\square$

From Theorem 1 and Lemma 4, we obtain

Theorem 2

There exists a constant $C_3$ depending only on $\chi (X)$ such that

$$\begin{aligned} ||\Phi (\rho )||_1&\le \frac{\pi |\chi (X)|}{2} e^{2 d_{\mathrm{{P}}_1} (\sigma , \rho )} + C_3 \\&\le \frac{\pi |\chi (X)|}{2} e^{2 d_\mathrm{{L}} (\sigma , \rho )} +C_3 \\&\le \frac{\pi |\chi (X)|}{2} e^{2 d_\mathrm{{T}} (\sigma , \rho )} +C_3 \end{aligned}$$

holds for any two points $[\sigma ]$ and $[\rho ]$ in $T(X)$.

For any $[\sigma ]$ and $[\rho ]$ in $T(X)$,

$$\begin{aligned} ||\Phi (\rho )||_1 \le C_4 (\sigma ) \pi |\chi (X)| e^{4 d_{\mathrm{{P}}_2} (\sigma , \rho )} +C_5, \end{aligned}$$

where $C_4 (\sigma )$ is a constant depending only on $[\sigma ]$ and $C_5$ is a constant depending only on $\chi (X)$.