[Paper Review] Plurisubharmonicity and geodesic convexity of energy function on Teichmüller space

Инканг Ким, Xueyuan Wan+1 more|arXiv (Cornell University)|Sep 1, 2018

Analytic and geometric function theory17 references4 citations

TL;DR

This paper establishes that the logarithm of the energy functional of harmonic maps from a Riemannian manifold to hyperbolic surfaces in Teichmüller space is strictly plurisubharmonic, providing a new proof of the Steinness of Teichmüller space. It further proves strict convexity of the energy function along Weil-Petersson geodesics and derives precise second variation formulas, including for geodesic length functions, and shows that $ E(t)^c $ is strictly convex for $ c > 5/6 $ and convex for $ c = 5/6 $.

ABSTRACT

Let $π:\mc{X} o \mc{T}$ be Teichmüller curve over Teichmüller space $\mc{T}$, such that the fiber $\mc{X}_z=π^{-1}(z)$ is exactly the Riemann surface given by the complex structure $z\in \mc{T}$. For a fixed Riemannian manifold $M$ and a continuous map $u_0: M o \mc{X}_{z_0}$, let $E(z)$ denote the energy function of the harmonic map $u(z):M o \mc{X}_z$ homotopic to $u_0$, $z\in \mathcal T$. We obtain the first and the second variations of the energy function $E(z)$, and show that $\log E(z)$ is strictly plurisubharmonic on Teichmüller space, from which we give a new proof on the Steinness of Teichmüller space. We also obtain a precise formula on the second variation of $E^{1/2}$ if $\dim M=1$. In particular, we get the formula of Axelsson-Schumacher on the second variation of the geodesic length function. We give also a simple and corrected proof for the theorem of Yamada, the convexity of energy function $E(t)$ along Weil-Petersson geodesics. As an application we show that $E(t)^c$ is also strictly convex for $c>5/6$ and convex for $c=5/6$ along Weil-Petersson geodesics. We also reprove a Kerckhoff's theorem which is a positive answer to the Nielsen realization problem.

Motivation & Objective

• Establish the plurisubharmonicity of the logarithm of the energy functional on Teichmüller space to provide a new proof of its Stein property.
• Derive precise first and second variation formulas for the energy functional $ E(z) $ of harmonic maps in the Teichmüller space setting.
• Prove strict convexity of $ E(t)^c $ along Weil-Petersson geodesics for $ c > 5/6 $, with convexity at $ c = 5/6 $, extending known results on geodesic length functions.
• Re-derive and correct the second variation formula for geodesic length functions via harmonic map energy methods.
• Provide a simplified and corrected proof of Yamada’s theorem on energy convexity along Weil-Petersson geodesics and reprove Kerckhoff’s Nielsen realization theorem using energy functionals.

Proposed method

• Use the Teichmüller curve $ ho: ilde{ ho} o ilde{ ho} $ to parametrize Riemann surfaces over Teichmüller space $ ilde{ ho} $, with fibers $ ilde{ ho}_z $ carrying hyperbolic metrics.
• Apply the theory of harmonic maps from a fixed Riemannian manifold $ (M^n, g) $ to the fiber $ ilde{ ho}_z $, ensuring existence and uniqueness of harmonic representatives homotopic to a fixed map $ u_0 $.
• Compute the first and second variations of the energy functional $ E(z) = \frac{1}{2} \int_M |du(z)|^2 d\mu_g $ using infinitesimal deformations of the complex structure and the associated Beltrami differentials.
• Derive the second variation formula involving the trace of the pullback metric and the Beltrami differential $ q $, showing that $ \delta^2 E(u_0)(W_0, W_0) \leq \int_M \frac{|q|^2}{\phi_0^2} \text{Tr}_g(u_0^* \Phi_0) d\mu_g $.
• Establish positivity of the second variation by combining curvature estimates and trace inequalities, proving $ \frac{d^2}{dt^2} E(t) \geq \int_M \frac{|q|^2}{3\phi_0^2} \text{Tr}_g(u_0^* \Phi_0) d\mu_g > 0 $.
• Use the second variation result to analyze $ \frac{d^2}{dt^2} E(t)^c $, deriving the condition $ c > 5/6 $ for strict convexity via a quadratic inequality involving $ \left( \frac{dE}{dt} \right)^2 \leq 6E \frac{d^2E}{dt^2} $.

Experimental results

Research questions

• RQ1 Is the logarithm of the energy functional $ \log E(z) $ strictly plurisubharmonic on Teichmüller space?
• RQ2 Does the energy functional $ E(t) $ remain strictly convex along Weil-Petersson geodesics, and for which powers $ c $ is $ E(t)^c $ convex or strictly convex?
• RQ3 What is the precise second variation formula for $ E^{1/2}(t) $ when the domain manifold has dimension one?
• RQ4 Can the second variation of the geodesic length function be recovered from the energy functional framework?
• RQ5 Can the Nielsen realization problem be re-proven using the convexity and properness of energy functionals on Teichmüller space?

Key findings

• The logarithm of the energy functional $ \log E(z) $ is strictly plurisubharmonic on Teichmüller space, which provides a new proof of the Stein property of Teichmüller space.
• The second variation of the energy functional $ E(t) $ along a Weil-Petersson geodesic is strictly positive, confirming Yamada’s result on strict convexity of $ E(t) $.
• The function $ E(t)^c $ is strictly convex along Weil-Petersson geodesics for $ c > 5/6 $, and convex for $ c = 5/6 $, with the threshold $ 5/6 $ arising from a quadratic inequality involving the first and second derivatives.
• By specializing to geodesic curves (where energy is the square of length), the paper recovers and corrects the second variation formula of Axelsson and Schumacher for the geodesic length function.
• The sum of energy functionals over a filling family of curves is proper and strictly convex, leading to a unique minimum point invariant under the group action, thus providing a new proof of Kerckhoff’s Nielsen realization theorem.
• The second variation of $ E^{1/2}(t) $ is derived explicitly for $ \dim M = 1 $, yielding a precise formula that recovers the known second variation of the geodesic length function.