Isometric embedding with nonnegative Gauss curvature under the graph setting

TL;DR

This paper investigates the regularity of isometric embeddings of 2-dimensional disks with nonnegative curvature into three-dimensional space, establishing optimal regularity results under a graph representation assumption.

Contribution

It proves that such embeddings are C2,1 regular near a point, under the graph form assumption, extending understanding of embedding regularity for nonnegatively curved surfaces.

Findings

Embeddings are C2,1 regular near a point.

Optimal regularity is achieved, matching known examples.

Results apply to surfaces with nonnegative Gauss curvature in the graph setting.

Abstract

We study the regularity of the isometric embedding X: (B(O,r),g) -> (R3, gcan) of a 2-ball with nonnegatively curved C4 metric into R3. Under the assumption that X can be expressed in the graph form, we show X is C2,1 near P, which is optimal by Iaia's example.