# Towards a constrained Willmore conjecture

## TL;DR

This paper reviews the constrained Willmore problem, discusses related conjectures, and explores their implications for understanding the Willmore functional on conformal immersions of compact surfaces.

## Contribution

It provides an overview of the constrained Willmore problem and proposes conjectures that could deepen understanding of the Willmore functional.

## Key findings

- Discussion of partial results and numerical experiments
- Formulation of new conjectures in the constrained Willmore problem
- Potential implications for conformal geometry and surface theory

## Abstract

We give an overview of the constrained Willmore problem and address some conjectures arising from partial results and numerical experiments. Ramifications of these conjectures would lead to a deeper understanding of the Willmore functional over conformal immersions from compact surfaces.