Steklov eigenvalues, homogenization and free boundary minimal surfaces

Abstract: It has been known since classical antiquity that disks have the largest area among planar figures of prescribed perimeter. Nevertheless, a rigorous proof was only given around the end of the 19th century. During the 20th century, area and perimeter were replaced by many other analytic and geometric quantities, and the geometric setting has been vastly enlarged. In this talk we will be interested in two such isoperimetric-type problems:

(A) Free boundary minimal surfaces The minimization of area for surfaces in balls, with their boundary that are constrained to live on the boundary sphere (free boundary minimal surfaces).

B) Isoperimetric problem for Steklov eigenvalues The maximization of the spectral gap of Dirichlet-to-Neumann operators for surfaces with prescribed perimeter.

For domains in the unit sphere and planar domains, I will describe the complete solution of problem (B). It is based on the theory of homogenization by perforation, a topic which comes from applied and industrial mathematics. Then, using work of Fraser and Schoen, I will show how this solution leads to the construction of new free boundary minimal surfaces in the unit 3-ball that have area larger than was previously thought possible.

This talk is based on joint work with Antoine Henrot, Mikhail Karpukhin and Jean Lagacé.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

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CRM - Séminaire du CIRGET / Géométrie et Topologie

Series comments: Hybrid seminar of geometry and topology. Laboratory : CIRGET - www.cirget.uqam.ca The homepage of the seminar is www.cirget.uqam.ca/fr/seminaires.html

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