Plurisuperharmonicity of reciprocal energy function on Teichmüller space and Weil-Petersson metric

Abstract

We consider harmonic maps u(z):Xz→N in a fixed homotopy class from Riemann surfaces Xz of genus g≥2 varying in the Teichmüller space 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 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.,(0.1)−1∂∂¯log⁡E(z)=ωWP2π(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 {Xz}z∈T in a fixed homotopy class. If u(z) is holomorphic or anti-holomorphic, then (0.1) is also proved.