Unknottedness of free boundary minimal surfaces and self-shrinkers

TL;DR

This paper proves that certain free boundary minimal surfaces and self-shrinkers in three-dimensional spaces are unknotted, meaning they are smoothly isotopic if they share the same boundary graph or graph at infinity, respectively.

Contribution

It introduces the concepts of boundary graph and graph at infinity, and establishes unknottedness results for these surfaces in specified geometric contexts.

Findings

Surfaces with the same boundary graph are smoothly isotopic.

Self-shrinkers with the same graph at infinity are smoothly isotopic.

The paper provides new tools for understanding the topology of minimal surfaces.

Abstract

We study unknottedness for free boundary minimal surfaces in a three-dimensional Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary, and for self-shrinkers in the three-dimensional Euclidean space. For doing so, we introduce the concepts of boundary graph for free boundary minimal surfaces and of graph at infinity for self-shrinkers. We prove that these surfaces are unknotted in the sense that any two such surfaces with isomorphic boundary graph or graph at infinity are smoothly isotopic.