Calculus of Variations and Geometric Measure Theory

L. Ambrozio - R. Buzano - A. Carlotto - B. Sharp

Bubbling analysis and geometric convergence results for free boundary minimal surfaces

created by muller on 21 Jul 2018
modified on 13 Jun 2022


Published Paper

Inserted: 21 jul 2018
Last Updated: 13 jun 2022

Journal: Journal de l'École Polytechnique - Mathématiques
Volume: 6
Year: 2019
Doi: 10.5802/jep.102

ArXiv: 1807.00632 PDF


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.