2018/09/01 by Инканг Ким, Xueyuan Wan, Kim, Inkang +3 · 1 citation
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1809.00255
openalex publication_date 2018/09/01 · openalex created_date 2018/09/27 · openalex updated_date 2026/07/28
Let π:\mcX→ \mcT be Teichmüller curve over Teichmüller space \mcT, such that the fiber \mcXz=π-1(z) is exactly the Riemann surface given by the complex structure z∈ \mcT. For a fixed Riemannian manifold M and a continuous map u0: M→ \mcXz0, let E(z) denote the energy function of the harmonic map u(z):M→ \mcXz homotopic to u0, z∈ \mathcal T. We obtain the first and the second variations of the energy function E(z), and show that log E(z) is strictly plurisubharmonic on Teichmüller space, from which we give a new proof on the Steinness of Teichmüller space. We also obtain a precise formula on the second variation of E1/2 if dim M=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>5/6 and convex for c=5/6 along Weil-Petersson geodesics. We also reprove a Kerckhoff's theorem which is a positive answer to the Nielsen realization problem.