Free boundary minimal surfaces in the unit ball : recent advances and open questions
arXiv:1907.05053
Abstract
In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary minimal surfaces. Moreover, we describe estimates for the first Steklov eigenvalue on such free boundary minimal surfaces and various smooth compactness results. Finally, we mention some sharp area bounds for free boundary minimal submanifolds and related questions.
arXiv commentrevised version with new references updated, to appear in Proceedings of the first annual meeting of the ICCM
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Cited by in corpus (9)
- Free boundary minimal surfaces with connected boundary and arbitrary genus
- Existence of infinitely many free boundary minimal hypersurfaces
- From Steklov to Laplace: free boundary minimal surfaces with many boundary components
- Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
- Disks area-minimizing in mean convex Riemannian -manifolds
- A Note on Free Boundary Hypersurfaces in Space Forms Balls
- Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary
- A Note On Free Boundary Minimal Annulus
- Free-Boundary Minimal Surfaces of Constant Kahler Angle in Complex Space Forms