Closed surfaces with bounds on their Willmore energy
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 605-634.

The Willmore energy of a closed surface in n is the integral of its squared mean curvature, and is invariant under Möbius transformations of n . We show that any torus in 3 with energy at most 8π-δ 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 δ>0. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus p1 in 3 or 4 , assuming suitable energy bounds which are sharp for n=3. Moreover, the conformal type is controlled in terms of the energy bounds.

Published online:
Classification: 53A05, 53A30, 53C21, 49Q15
Kuwert, Ernst 1; Schätzle, Reiner 2

1 Mathematisches Institut Universität Freiburg Eckerstraße 1 D-79104 Freiburg
2 Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen
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Kuwert, Ernst; Schätzle, Reiner. Closed surfaces with bounds on their Willmore energy. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 605-634. http://archive.numdam.org/item/ASNSP_2012_5_11_3_605_0/

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