Hermitian Curvature and Plurisubharmonicity of Energy on Teichm\"uller Space

TL;DR

This paper proves that the energy function of harmonic maps from a Riemann surface to a Hermitian non-positively curved manifold is plurisubharmonic on Teichmüller space, with conditions for vanishing Hessian in strictly negative curvature cases.

Contribution

It establishes the plurisubharmonicity of the energy function on Teichmüller space and characterizes the directions of vanishing complex Hessian when the target has strictly negative Hermitian curvature.

Findings

Energy function is plurisubharmonic on Teichmüller space.

Characterization of directions with vanishing Hessian under negative curvature.

Provides geometric insights into harmonic maps and Teichmüller theory.

Abstract

Let $M$ be a closed Riemann surface, $N$ a Riemannian manifold of Hermitian non-positive curvature, $f:M\to N$ a continuous map, and $E$ the function on the Teichm\"uller space of $M$ that assigns to a complex structure on $M$ the energy of the harmonic map homotopic to $f$. We show that $E$ is a plurisubharmonic function on the Teichm\"uller space of $M$. If $N$ has strictly negative Hermitian curvature, we characterize the directions in which the complex Hessian of $E$ vanishes.