## 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

[

BibTeX]

*preprint*

**Inserted:** 21 jul 2018

**Year:** 2018

**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.