Connected sums of constant mean curvature surfaces in Euclidean 3 space

TL;DR

This paper proves a gluing theorem for constant mean curvature surfaces in Euclidean 3-space, allowing the construction of new connected sum surfaces from two given nondegenerate surfaces with parallel tangent planes.

Contribution

It introduces a general gluing method for creating connected sum constant mean curvature surfaces, extending to compact and noncompact cases.

Findings

New connected sum CMC surfaces constructed near given configurations

The resulting surfaces are nondegenerate for generic parameters

The method applies to both compact with boundary and complete noncompact surfaces

Abstract

We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. Here nondegeneracy refers to the invertibility of the linearized mean curvature operator. This paper treats the simplest context for our result namely when the surfaces are compact with nonempty boundary, however the construction applies in the complete, noncompact setting as well. The surfaces we produce here are nondegenerate for generic choices of the free parameters in the construction.