# Conformal Gauss Map Geometry and Application to Willmore Surfaces in Model Spaces

## TL;DR

This paper provides a comprehensive study of the conformal Gauss map, exploring its properties and applications to Willmore surfaces, including new characterizations of minimal and CMC surfaces.

## Contribution

It offers a detailed analysis of the conformal Gauss map and introduces novel characterizations of minimal and CMC surfaces based on its behavior.

## Key findings

- Characterizations of minimal surfaces via conformal Gauss map
- Characterizations of constant mean curvature surfaces
- Enhanced understanding of Willmore surfaces

## Abstract

In this paper we make a detailed and self-contained study of the conformalGauss map. Then, starting from the seminal work of R. Bryant and the notion of conformal Gauss map, we recover many fundamental properties of Willmore surfaces. We also get new results like some characterizations of minimal and constant meancurvature (CMC) surfaces in term of their conformal Gauss map behavior.