Navigating the Space of Symmetric CMC Surfaces

TL;DR

This paper introduces a flow on spectral data for symmetric CMC surfaces in the 3-sphere, enabling the construction of higher genus surfaces and connecting spectral data evolution to classical minimal surface deformations.

Contribution

It develops a novel flow on spectral data that alters topology while preserving key geometric conditions, and establishes short-term existence near CMC tori spectral data.

Findings

Flow produces closed CMC surfaces of higher genus at rational times.

Flow near Clifford torus spectral data exists for short times.

Flow relates to classical minimal surface deformations in Lawson's construction.

Abstract

In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. For rational times we obtain closed (possibly branched) connected CMC surfaces of higher genus. We prove the short time existence of this flow near the spectral data of (a class of) CMC tori. In particular we prove that flowing the spectral data for the Clifford torus is equivalent to the flow of Plateau solutions by varying the angle of the fundamental piece in Lawson's construction for the minimal surfaces $\xi_{g,1}.$