Closed surfaces with bounds on their Willmore energy

Abstract

The Willmore energy of a closed surface in Rn is the integral of its squared mean curvature, and is invariant under Mobius transformations of Rn . We show that any torus in R3 with energy at most 8π − δ has a representative under the Mobius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on δ > 0. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus p ≥ 1 in R3 or R4, assuming suitable energy bounds which are sharp for n = 3. Moreover, the conformal type is controlled in terms of the energy bounds. Mathematics Subject Classification (2010): 53A05 (primary); 53A30, 53C21, 49Q15 (secondary).