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 logE(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∂∂¯logE(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.
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