Plurisuperharmonicity of reciprocal energy function on Teichmuller space and Weil-Petersson metrics

Abstract

We consider harmonic maps$u(z):{\mathcal{X}}_z\to N$ in a fixed homotopy class from Riemann surfaces ${\mathcal{X}}_z$ of genus $g\ge 2$ varying in the Teichmü{}ller space $\mathcal{T}$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a function on $\mathcal{T}$ and we study its first and the second variations. We prove that the reciprocal energy function $E(z{)}^{-1}$ is plurisuperharmonic on Teichmüller space. We also obtain the (strict) plurisubharmonicity of $\log E(z)$ and $E(z)$. As an application, we get the following relationship between the second variation of logarithmic energy function and the Weil-Petersson metric if the harmonic map $u(z)$ is holomorphic or anti-holomorphic and totally geodesic, i.e., $$ \sqrt{-1}\p\b{\p}\log E(z)=\frac{ω_{WP}}{2π(g-1)}. $$ We consider also the energy function $E(z)$ associated to the harmonic maps from a fixed compact Kähler manifold $M$ to Riemann surfaces ${\mathcal{X}_z\}_{z\in\mathcal{T}}$ in a fixed homotopy class. If $u(z)$ is holomorphic or anti-holomorphic, then the above equation is also proved.

27 pages

Cited by in corpus (1)

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