Curvature estimates for surfaces with bounded mean curvature

TL;DR

This paper establishes bounds on the curvature of embedded surfaces with controlled mean curvature, extending classical estimates for minimal surfaces to more general cases with small mean curvature in certain Sobolev spaces.

Contribution

It proves that for surfaces with bounded mean curvature in a specific Sobolev norm, the second fundamental form is bounded at interior points, generalizing known minimal surface estimates.

Findings

Bound on |A| under small mean curvature in W^{1,p} norm

Extension of classical minimal surface estimates

Curvature control for surfaces with bounded L^2 norm of |A|

Abstract

Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for minimal surfaces.