# Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of revolution

## TL;DR

This paper derives explicit formulas for all revolution Willmore surfaces, explores their boundary value problems, and investigates symmetry properties, including symmetry breaking and conditions for symmetry preservation.

## Contribution

It provides the first explicit formulas for revolution Willmore surfaces and analyzes symmetry behavior under boundary conditions.

## Key findings

- Explicit formulas for all revolution Willmore surfaces
- Symmetry breaking can occur under symmetric boundary conditions
- A symmetry result for a subclass of surfaces with symmetric boundary data

## Abstract

In this paper we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addition we prove a symmetry result for a subclass of Willmore surfaces satisfying symmetric Dirichlet boundary data.