The rigidity of embedded constant mean curvature surfaces

TL;DR

This paper investigates the rigidity properties of complete, embedded constant mean curvature surfaces in three-dimensional space, establishing conditions under which intrinsic symmetries extend to ambient space symmetries.

Contribution

It proves that finite genus embedded constant mean curvature surfaces have their intrinsic isometries extend to ambient isometries or relate to a subgroup of the ambient isometry group.

Findings

Intrinsic isometries extend to ambient isometries for finite genus surfaces.

The isometry group contains an index two subgroup extending to R^3.

Rigidity properties depend on the genus of the surface.

Abstract

We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its isometry group contains an index two subgroup of isometries that extend.