Minimizers of the Willmore functional under fixed conformal class

Abstract

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into $\mathbb{R}^n, n = 3, 4$, if there is one conformal immersion with Willmore energy smaller than a certain bound $\mathcal{W}_{n,p}$ depending on codimension and genus $p$ of the Riemann surface. For tori in codimension 1, we know $\mathcal{W}_{3,1} = 8\pi$.