Free boundary minimal surfaces in the unit ball : recent advances and open questions

TL;DR

This survey reviews recent progress and open questions in the study of free boundary minimal surfaces within the unit ball, highlighting new constructions, uniqueness results, index estimates, and eigenvalue bounds.

Contribution

It compiles recent techniques, results, and conjectures in the field, emphasizing the development of new examples and understanding of free boundary minimal surfaces.

Findings

Construction methods for embedded free boundary minimal surfaces.

Results on the uniqueness of free boundary minimal disks and the critical catenoid.

Morse index estimates and Steklov eigenvalue bounds for these surfaces.

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.