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Connections Workshop: New Frontiers in Curvature & Special Geometric Structures and Analysis: Relating the index and the topology of (free boundary) minimal surfaces

Presenter
August 23, 2024
SLMath
Keywords:
  • Riemannian geometry
  • curvature
  • geometric flow
  • special holonomy
  • Minimal surface
  • general relativity
  • K¨ahler geometry
MSC:
  • 32-XX - Several complex variables and analytic spaces
  • 53-XX - Differential geometry
  • 58-XX - Global analysis analysis on manifolds
  • 83-XX - Relativity and gravitational theory
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Abstract
In this talk, I will discuss existing results and open problems about estimating the Morse index of a (free boundary) minimal surface from below by a function of its topology. I will mostly focus on results proving that the Morse index of a free boundary minimal surface in a three-dimensional Riemannian manifold grows linearly with the product of its area and its topology. This is joint work with Santiago Cordero-Misteli, and it is inspired by Antoine Song's result in the closed case.