Plurisuperharmonicity of reciprocal energy function on Teichmüller space and Weil-Petersson metric
Journal article, 2020

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., [Formula presented] 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.

Weil-Petersson metric

Energy function

Teichmüller space

Harmonic map


Inkang Kim

Korea Institute for Advanced Study

Xueyuan Wan

Korea Institute for Advanced Study

Genkai Zhang

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Journal des Mathematiques Pures et Appliquees

0021-7824 (ISSN)

Vol. 141 316-341

Subject Categories

Algebra and Logic


Mathematical Analysis



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