# The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

## TL;DR

This paper studies the 'area blow-up' set of bounded mean curvature submanifolds under anisotropic energies, showing it is empty in many cases and deriving boundary curvature estimates for stable anisotropic minimal surfaces.

## Contribution

It extends White's ideas to anisotropic settings, proving the emptiness of the blow-up set and establishing boundary curvature bounds for stable anisotropic minimal surfaces.

## Key findings

- The 'area blow-up' set has bounded anisotropic mean curvature in viscosity sense.
- The blow-up set is empty in various scenarios.
- Boundary curvature estimates are obtained for 2D stable anisotropic minimal surfaces.

## Abstract

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of (Invent. Math., 1987).