# On the umbilic set of immersed surfaces in three-dimensional space forms

## TL;DR

This paper proves that under certain conditions on mean curvature, the set of umbilic points on an immersed surface in 3D space forms has positive measure, extending classical results like Hopf's theorem for spheres.

## Contribution

It generalizes the Hopf theorem by showing the umbilic set has positive measure under specific mean curvature assumptions for immersed surfaces.

## Key findings

- Umbilic set has positive measure under certain conditions.
- Generalization of Hopf theorem for immersed spheres.
- Conditions on mean curvature are crucial for results.

## Abstract

We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.