Robert Kusner: Minimal Surfaces in Round Spheres and Balls

Robert Kusner (UMass)

Title: Minimal Surfaces in Round Spheres and Balls

Abstract:

A deep connection between extremal eigenvalue problems and minimal surfaces in round spheres or balls has emerged over the past decades, stemming from work of Nadirashvili and Fraser-Schoen.  We use this to determine geometric properties of these surfaces, like a uniqueness theorem for embedded free boundary minimal annuli with antipodal symmetry, by showing the first eigenspace for many such surfaces coincides with the span of the ambient coordinate functions (generalizing the Yau and Fraser-Li conjectures).   We also develop an equivariant version of eigenvalue optimization with sharp a priori   eigenspace dimension bounds, letting us construct free boundary minimal surfaces of every topological type embedded in the round 3-ball, and many more closed minimal surfaces embedded in the round 3-sphere.  

[Based on joint projects with Misha Karpukhin, Peter McGrath and Daniel Stern]