Bubbling analysis and geometric convergence results for free boundary minimal surfaces

TL;DR

This paper studies the behavior of free boundary minimal hypersurfaces with bounded index and volume, establishing a quantization identity for total curvature and analyzing convergence and degeneration phenomena, especially in three dimensions.

Contribution

It introduces a detailed blow-up analysis near curvature concentration points and derives a general quantization identity applicable in dimensions less than eight.

Findings

Quantization identity for total curvature in ambient dimension less than eight

Classification of degenerations for free boundary minimal surfaces in certain 3D domains

Unconditional convergence results for most topologies in strictly mean convex domains

Abstract

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in $\mathbb{R}^3$ unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.