2012/09/30 by Ailana Fraser, Richard Schoen · 4 citations
Mathematics · #math.DG #msc:35P15 #msc:53A10
paper · pdf · doi:10.1007/s00222-015-0604-x
published as Invent. Math. 203 (2016), no. 3, 823-890 · 52 pages. Final version that appeared in Invent. Math.: Presentation of Proposition 4.3 improved, other minor edits throughout
arxiv created 2017/11/13 · arxiv updated 2017/11/15
We prove existence and regularity of metrics on a surface with boundary which maximize sigma1 L where sigma1 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 Bn 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 B3. We also show that the unique solution on the Mobius band is achieved by an explicit S1 invariant embedding in B4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma1 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.