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Topological control for min-max free boundary minimal surfaces

Giada Franz, Mario B. Schulz

TL;DR

This work develops sharp topological control for free boundary minimal surfaces produced by min-max methods in 3D manifolds with mean convex boundary. It introduces genus and boundary complexities $\mathfrak{g}$ and $\mathfrak{b}$ and proves that, under varifold convergence, $\beta_1$ of the limit is bounded below by the liminf of $\beta_1$ along the sequence, while, in the orientable case, $\mathfrak{g}$ and $\mathfrak{b}$ satisfy analogous semicontinuity and their sum is nonincreasing. Central to the arguments are Simon’s lifting lemma and a new free-boundary variant, together with a controlled surgery procedure that preserves orientability and reduces topology where appropriate. These tools enable concrete applications, including a variational construction of a free boundary minimal trinoid in the unit ball, and refinement of index bounds for min-max surfaces. Overall, the results provide robust topological constraints for free boundary min-max constructions and broaden the scope of explicit, low-genus examples in convex and symmetric ambient spaces.

Abstract

We establish general bounds on the topology of free boundary minimal surfaces obtained via min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We prove that the first Betti number is lower semicontinuous along min-max sequences converging in the sense of varifolds to free boundary minimal surfaces. In the orientable case, we obtain an even stronger result which implies that if the number of boundary components increases in the varifold limit, then the genus decreases at least as much. We also present several compelling applications, such as the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.

Topological control for min-max free boundary minimal surfaces

TL;DR

This work develops sharp topological control for free boundary minimal surfaces produced by min-max methods in 3D manifolds with mean convex boundary. It introduces genus and boundary complexities and and proves that, under varifold convergence, of the limit is bounded below by the liminf of along the sequence, while, in the orientable case, and satisfy analogous semicontinuity and their sum is nonincreasing. Central to the arguments are Simon’s lifting lemma and a new free-boundary variant, together with a controlled surgery procedure that preserves orientability and reduces topology where appropriate. These tools enable concrete applications, including a variational construction of a free boundary minimal trinoid in the unit ball, and refinement of index bounds for min-max surfaces. Overall, the results provide robust topological constraints for free boundary min-max constructions and broaden the scope of explicit, low-genus examples in convex and symmetric ambient spaces.

Abstract

We establish general bounds on the topology of free boundary minimal surfaces obtained via min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We prove that the first Betti number is lower semicontinuous along min-max sequences converging in the sense of varifolds to free boundary minimal surfaces. In the orientable case, we obtain an even stronger result which implies that if the number of boundary components increases in the varifold limit, then the genus decreases at least as much. We also present several compelling applications, such as the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.
Paper Structure (15 sections, 19 theorems, 48 equations, 4 figures)

This paper contains 15 sections, 19 theorems, 48 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and convention
  3. Equivariant min-max theory
  4. Main results
  5. Acknowledgements.
  6. Direct applications
  7. Discs.
  8. Connected boundary and arbitrary genus.
  9. Index five.
  10. Surgery
  11. Topological lower semicontinuity
  12. Lifting lemmata
  13. Proof of the main results
  14. Free boundary minimal surfaces in the unit ball with genus zero
  15. Relating topological invariants

Key Result

Theorem 1.4

Let $M$ be a three-dimensional Riemannian manifold with strictly mean convex boundary and let $G$ be a finite group of orientation-preserving isometries of $M$. Let $\{\Sigma_t\}_{t\in[0,1]^n}$ be a $G$-sweepout of $M$. If the min-max width $W_\Pi$ of its $G$-saturation satisfies then there exists a min-max sequence $\{\Sigma^j\}_{j\in\mathbb{N}}$ of (smooth) $G$-equivariant surfaces converging i

Figures (4)

  • Figure 1: Left pair: Cutting away a neck. Right pair: Cutting away a half-neck.
  • Figure 2: Diagram of the varifold convergence in a small ball $B$ described in Lemma \ref{['lem:LocalPictureConv']}.
  • Figure 3: Small balls $B^{\alpha}\vcentcolon=B_\rho(x_\alpha)\subset M$ around points $x_0,\ldots,x_N$ on the edge $\gamma$ of the tree $T$.
  • Figure 4: Successive stages of the sweepout in the case $n=3$.

Theorems & Definitions (55)

  • Definition 1.1: $G$-sweepout
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4: Equivariant min-max
  • Remark 1.5
  • Definition 1.6: Topological complexity
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 45 more