Plurisubharmonicity and Geodesic Convexity of Energy Function on Teichmuller Space

Journal article, 2022

Let π : X → T be Teichmüller curve over Teichmüller space T, such that the fiber Xz = π−1(z) is exactly the Riemann surface given by the complex structure z ∈ T . For a fixed Riemannian manifold M and a continuous map u0 : M → Xz0, let E(z) denote the energy function of the harmonic map u(z) : M → Xz homotopic to u0, z ∈ T . We obtain the first and the second variations of the energy function E(z), and show that logE(z)is strictly plurisubharmonic on Teichmüller space, and that both E(z) and logE(z) are plurisubharmonic exhausting functions. We also obtain a precise formula on the second variation of E1/2 if dimM = 1. In particular, we get the formula of Axelsson-Schumacher on the second variation of the geodesic length function. We give also a simple and corrected proof for the theorem of Yamada, the convexity of energy function E(t) along Weil-Petersson geodesics. As an application we show that E(t)c is also strictly convex for c > 65 and convex for c = 65 along Weil-Petersson geodesics. We also re-prove a Kerckhoff’s theorem, which is a positive answer to the Nielsen realization problem.