Minimizers of the Willmore functional under fixed conformal class

TL;DR

This paper proves the existence of smooth minimizers of the Willmore energy within fixed conformal classes of Riemann surfaces, establishing bounds and explicit values for tori in three-dimensional space.

Contribution

It demonstrates the existence of smooth minimizers of the Willmore functional under fixed conformal class constraints, with explicit bounds for specific cases.

Findings

Existence of smooth minimizers under certain energy bounds.

Explicit value of the Willmore energy bound for tori in R^3 as 8π.

Establishment of conditions ensuring minimizer existence.

Abstract

We consider the problem of minimizing the Willmore energy in the class of conformal immersions of a given closed, genus p Riemann surface into R^n for n=3,4. We prove existence of a smooth minimizer, provided that the infimum is below a certain bound ${\cal W}(n,p)$. For tori in R^3 we have explicitely ${\cal W}(3,1) = 8\pi$.