Closed surfaces with bounds on their Willmore energy

Abstract

The Willmore energy of a closed surface in R^n is the integral of its squared mean curvature, and is invariant uner Möbius transformations of R^n. We show that any torus in R^3 with energy at most $8\pi -delta$ has a representative under the Möbius action, for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on $delta>0$. An analogous estimate is also obtained for surfaces of fixed genus $p\ge 1$ in R^3 or R^4, assuming suitable energy bounds which are sharp for n=3. Moreover the conformal type is controlled in terms of the energy bounds.