Minimal surfaces in the three dimensional sphere with high symmetry

TL;DR

This paper constructs new symmetric embedded minimal surfaces in the 3-sphere using Lawson's theorem and Hopf fibration symmetries, revealing seven novel surfaces with high genus and exploring their symmetry group actions.

Contribution

It introduces seven new high-genus minimal surfaces in the 3-sphere and analyzes their symmetry properties, expanding the known catalog of such surfaces.

Findings

Seven new minimal surfaces with genera 9, 25, 49, 121, 121, 361, 841 identified.

Established relations between these surfaces and maximal extendable group actions.

Connected the constructed surfaces to known examples like the Clifford and Lawson's minimal surfaces.

Abstract

Using the Lawson's existence theorem of minimal surfaces and the symmetries of the Hopf fibration, we will construct symmetric embedded closed minimal surfaces in the three dimensional sphere. These surfaces contain the Clifford torus, the Lawson's minimal surfaces, and seven new minimal surfaces with genera 9, 25, 49, 121, 121, 361 and 841. We will also discuss the relation between such surfaces and the maximal extendable group actions on subsurfaces of the three dimensional sphere.