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Unknottedness of free boundary minimal surfaces and self-shrinkers

Sabine Chu, Giada Franz

TL;DR

The paper investigates unknottedness for free boundary minimal surfaces in manifolds with nonnegative Ricci curvature and convex boundary, and for self-shrinkers in R^3, by introducing boundary graphs and graphs at infinity. It proves that two such surfaces with the same genus and isomorphic boundary data are smoothly isotopic, reducing the problem to showing strong Heegaard splitting via a Frankel-property framework and universal-cover arguments. The main technical engine uses weighted metric arguments (including a Gaussian-type weight for self-shrinkers) to obtain π1-surjectivity in the complement, enabling isotopy to a model surface and, hence, unknottedness. The work also clarifies the structure of admissible boundary graphs (trees, with genus-zero cases yielding stars) and ends, and discusses open questions about which graphs can be realized, highlighting rigidity phenomena in the self-shrinker setting.

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.

Unknottedness of free boundary minimal surfaces and self-shrinkers

TL;DR

The paper investigates unknottedness for free boundary minimal surfaces in manifolds with nonnegative Ricci curvature and convex boundary, and for self-shrinkers in R^3, by introducing boundary graphs and graphs at infinity. It proves that two such surfaces with the same genus and isomorphic boundary data are smoothly isotopic, reducing the problem to showing strong Heegaard splitting via a Frankel-property framework and universal-cover arguments. The main technical engine uses weighted metric arguments (including a Gaussian-type weight for self-shrinkers) to obtain π1-surjectivity in the complement, enabling isotopy to a model surface and, hence, unknottedness. The work also clarifies the structure of admissible boundary graphs (trees, with genus-zero cases yielding stars) and ends, and discusses open questions about which graphs can be realized, highlighting rigidity phenomena in the self-shrinker setting.

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.

Paper Structure

This paper contains 7 sections, 7 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free boundary minimal surfaces
  3. Self-shrinkers
  4. Key ideas of the proofs
  5. Topological preliminaries
  6. Free boundary minimal surfaces are strong Heegaard splittings
  7. Self-shrinkers are strong Heegaard splittings

Key Result

Theorem 1.0

Let $\Sigma,\ \Sigma'$ be (smooth, compact, properly embedded) free boundary minimal surfaces in a compact three-dimensional Riemannian manifold $(M^3,g)$ with nonnegative Ricci curvature and strictly convex boundary. Assume that $\Sigma$ and $\Sigma'$ have the same genus and that their boundary gra

Figures (2)

  • Figure 1: Schematic representation of the free boundary minimal surfaces in KapouleasLi2017 (top row) and KarpukhinKusnerMcGrathStern2024 (bottom row), together with the arrangement of the boundary components on the sphere and the resulting boundary graphs. See SchulzGallery for more precise numerical simulations of these surfaces.
  • Figure 2: Schematical representation of a model surface with prescribed genus and boundary graph.

Theorems & Definitions (20)

  • Theorem 1.0
  • Theorem 1.0
  • Definition 2.1: cf. Frohman1992*Section 2
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: cf. Frohman1992*Section 1
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7: Model surface
  • ...and 10 more