Closed surfaces with bounds on their Willmore energy

Kuwert, Ernst; Schätzle, Reiner

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 3, pp. 605-634.

The Willmore energy of a closed surface in ${ℝ}^n$ is the integral of its squared mean curvature, and is invariant under 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 closed, orientable 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.

Classification : 53A05, 53A30, 53C21, 49Q15

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