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Differential Geometry Seminar
Index and total curvature of minimal surfaces in noncompact symmetric spaces and wild harmonic bundles
Index and total curvature of minimal surfaces in noncompact symmetric spaces and wild harmonic bundles
Organizers
Speaker
Qiongling Li
Time
Tuesday, November 26, 2024 3:00 PM - 4:00 PM
Venue
A7-101
Online
Zoom 638 227 8222
(BIMSA)
Abstract
We prove two main theorems about equivariant minimal surfaces in arbitrary nonpositively curved symmetric spaces extending classical results on minimal surfaces in Euclidean space. First, we show that a complete equivariant branched immersed minimal surface in a nonpositively curved symmetric space of finite total curvature must be of finite Morse index. It is a generalization of the theorem by Fischer-Colbrie, Gulliver-Lawson, and Nayatani for complete minimal surfaces in Euclidean space. Secondly, we show that a complete equivariant minimal surface in a nonpositively curved symmetric space is of finite total curvature if and only if it arises from a wild harmonic bundle over a compact Riemann surface with finite punctures. Moreover, we deduce the Jorge-Meeks type formula of the total curvature and show it is an integer multiple of $2\pi/N$ for $N$ only depending on the symmetric space. It is a generalization of the theorem by Chern-Osserman for complete minimal surfaces in Euclidean n-space. This is joint work with Takuro Mochizuki (RIMS).