Structure theorems for embedded disks with mean curvature bounded in L^P

TL;DR

This paper proves that embedded disks with bounded mean curvature in L^P norm contain multi-valued graphs, extending known results from minimal surfaces to more general cases.

Contribution

It generalizes a classical result by Colding and Minicozzi to surfaces with non-zero mean curvature under L^P bounds.

Findings

Embedded disks with small L^P mean curvature contain multi-valued graphs.

The result applies after appropriate normalization of the surface.

Extends known minimal surface theory to broader class of surfaces.

Abstract

After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well known result of Colding and Minicozzi.