Plurisubharmonicity and Geodesic Convexity of Energy Function on Teichmuller Space
Artikel i vetenskaplig tidskrift, 2022

Let pi : X -> T be Teichmuller curve over Teichmuller space T, such that the fiber X-z = pi(-1) (z) is exactly the Riemann surface given by the complex structure z is an element of T. For a fixed Riemannian manifold M and a continuous map u(0): M -> X-z0, let E(z) denote the energy function of the harmonic map u(z) : M -> X-z homotopic to u(0), z is an element of T. We obtain the first and the second variations of the energy function E(z), and show that logE(z) is strictly plurisubharmonic on Teichmuller space, and that both E(z) and logE(z) are plurisubharmonic exhausting functions. We also obtain a precise formula on the second variation of E- (1/2) 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 re-prove a Kerckhoff's theorem, which is a positive answer to the Nielsen realization problem.

energy function

Harmonic map

Teichmtiller space

Weil-Petersson metric

Författare

Inkang Kim

Korea Institute for Advanced Study

Xueyuan Wan

Chongqing University of Technology

Genkai Zhang

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Göteborgs universitet

Indiana University Mathematics Journal

0022-2518 (ISSN)

Vol. 71 1 1-36

Representationer av Liegrupper. Harmonisk och komplex analys på symmetriska och lokalt symmetriska rum

Vetenskapsrådet (VR) (2018-03402), 2019-01-01 -- 2022-12-31.

Ämneskategorier

Algebra och logik

Geometri

Matematisk analys

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Senast uppdaterat

2022-03-29