Totally umbilical disks and applications to surfaces in three-dimensional homogeneous spaces

TL;DR

This paper establishes conditions under which disk-like surfaces are totally umbilical in certain geometric contexts, leading to new rigidity results for surfaces in space forms and homogeneous spaces.

Contribution

It provides new sufficient conditions for totally umbilical surfaces in the context of Coddazi pairs, extending known rigidity results to broader classes of spaces.

Findings

Derived new rigidity theorems for surfaces in space forms.

Established criteria for totally umbilical disks with piecewise smooth boundary.

Generalized previous results to homogeneous product spaces.

Abstract

Following ideas of Choe and Fernandez-do Carmo, we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for surfaces in space forms and in homogeneous product spaces that generalizes some known results.