Isometric deformations of isotropic surfaces

TL;DR

This paper extends a known finiteness result for minimal isometric immersions from compact non-simply-connected surfaces in 3-spheres to isotropic surfaces in spheres of any dimension, also addressing non-compact cases.

Contribution

It generalizes Ramanathan's finiteness theorem to isotropic surfaces in higher-dimensional spheres and includes non-compact surfaces in space forms.

Findings

Finite set of minimal isometric immersions for compact isotropic surfaces in spheres of any dimension

Extension of Ramanathan's result beyond 3-spheres

Addresses non-compact isotropic surfaces in space forms

Abstract

It was shown by Ramanathan \cite{R} that any compact oriented non-simply-connected minimal surface in the three-dimensional round sphere admits at most a finite set of pairwise noncongruent minimal isometric immersions. Here we show that this result extends to isotropic surfaces in spheres of arbitrary dimension. The case of non-compact isotropic surfaces in space forms is also addressed.