Minimizers of the Willmore functional under fixed conformal class

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 ${\cal W}(3,1) = 8\pi$.