# On Isolated Umbilic Points

## TL;DR

This paper constructs counter-examples to classical conjectures about umbilic points on surfaces, showing that certain bounds do not hold even for metrics close to flat, thus challenging existing geometric assumptions.

## Contribution

It explicitly constructs Riemannian metrics near Euclidean space that contain umbilic points contradicting Caratheodory's and Hamburger's conjectures.

## Key findings

- Counter-examples to Caratheodory's conjecture are constructed.
- Metrics with arbitrary umbilic index are demonstrated.
- Metrics can be arbitrarily close to flat Euclidean metrics.

## Abstract

Counter-examples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3-space. In particular, Riemannian metrics with a smooth strictly convex 2-sphere containing a single umbilic point are constructed explicitly, in contradiction with any direct extension of Caratheodory's conjecture. Additionally, a Riemannian metric with an embedded surface containing an isolated umbilic point of any index is presented, violating Hamburger's umbilic index bound. In both cases, it is shown that the metric can be made arbitrarily close to the flat metric. A short video explaining the motivation and results of this paper can be found at the following link: https://youtu.be/Wjja4PcMtxc