Sharp eigenvalue bounds and minimal surfaces in the ball

• 成果类型: Article
• 署名作者: Fraser, Ailana; Schoen, Richard
• 署名单位: University of British Columbia; Stanford University; University of California System; University of California Irvine
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0604-x
• 发表日期: 2016
• 页码: 823-890
• 关键词: 1st eigenvalue; immersions; 3-manifolds; laplacian

摘要

We prove existence and regularity of metrics on a surface with boundary which maximize sigma(1) L where sigma(1) is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B-n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B-3. We also show that the unique solution on the Mobius band is achieved by an explicit S-1 invariant embedding in B-4 as a free boundary surface, the critical Mbius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given byminimal embeddings in B-3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4 pi. We also prove multiplicity bounds on sigma(1) in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.