Connections Workshop: New Frontiers in Curvature & Special Geometric Structures and Analysis: Eigenvalue problems and free boundary minimal surfaces in spherical caps
Presenter
August 21, 2024
Keywords:
- Riemannian geometry
- curvature
- geometric flow
- special holonomy
- Minimal surface
- general relativity
- K¨ahler geometry
MSC:
- 32-XX - Several complex variables and analytic spaces {For infinite-\newline dimensional holomorphy
- see also 46G20
- 58B12}
- 53-XX - Differential geometry {For differential topology
- see 57Rxx
- for foundational questions of differentiable manifolds
- see 58Axx}
- 58-XX - Global analysis
- analysis on manifolds [See also 32Cxx
- 32Fxx
- 32Wxx
- 46-XX
- 53Cxx] {For nonlinear operators
- see 47Hxx
- for geometric integration theory
- see 49Q15}
- 83-XX - Relativity and gravitational theory
Abstract
In a joint work with Vanderson Lima (UFRGS, Brazil), we introduced a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk we will prove that each such functional is uniformly bounded from above, and we will characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we will determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we will prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.