# Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals

## TL;DR

This paper establishes a compactness theorem for volume-constrained almost-critical points of elliptic integrands, leading to new insights into critical points, local minimizers, and an Alexandrov-type theorem for crystalline isoperimetric problems.

## Contribution

It introduces a novel compactness theorem for almost-critical points measured in an integral sense, with applications to crystalline isoperimetric problems and elliptic energies.

## Key findings

- Compactness theorem for volume-constrained almost-critical points.
- Description of critical points and local minimizers with confinement.
- An Alexandrov-type theorem for crystalline isoperimetric problems.

## Abstract

A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main applications of the compactness theorem are discussed. First, we obtain a description of critical points/local minimizers of elliptic energies interacting with a confinement potential. Second, we prove an Alexandrov-type theorem for crystalline isoperimetric problems.