Isometric deformations of minimal surfaces in $S^{4}$

TL;DR

This paper investigates the classification of isometric minimal surfaces in the 4-sphere, showing that their moduli space is either finite or a circle, and establishing finiteness results for surfaces with nontrivial normal bundle.

Contribution

It characterizes the space of isometric minimal immersions in $S^{4}$ with the same normal curvature, revealing a dichotomy between finite sets and circles, and proves finiteness for surfaces with nontrivial normal bundle.

Findings

The space of isometric minimal immersions with the same normal curvature is either finite or a circle.

For compact minimal surfaces with nontrivial normal bundle, only finitely many noncongruent isometric immersions exist.

Provides a classification framework for minimal surfaces in $S^{4}$ based on their isometric deformations.

Abstract

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature function is, within congruences, either finite or a circle. Furthermore, we show that for any compact immersed minimal surface in $S^{4}$ with nontrivial normal bundle there are at most finitely many noncongruent immersed minimal surfaces in $S^{4}$ isometric to it with the same normal curvature function.