The Willmore energy of a closed surface in is the integral of its squared mean curvature, and is invariant under Möbius transformations of . We show that any torus in with energy at most 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 . An analogous estimate is also obtained for closed, orientable surfaces of fixed genus in or , assuming suitable energy bounds which are sharp for . Moreover, the conformal type is controlled in terms of the energy bounds.
@article{ASNSP_2012_5_11_3_605_0, author = {Kuwert, Ernst and Sch\"atzle, Reiner}, title = {Closed surfaces with bounds on their {Willmore} energy}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {605--634}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {3}, year = {2012}, mrnumber = {3059839}, zbl = {1260.53027}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2012_5_11_3_605_0/} }
TY - JOUR AU - Kuwert, Ernst AU - Schätzle, Reiner TI - Closed surfaces with bounds on their Willmore energy JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2012 SP - 605 EP - 634 VL - 11 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2012_5_11_3_605_0/ LA - en ID - ASNSP_2012_5_11_3_605_0 ER -
%0 Journal Article %A Kuwert, Ernst %A Schätzle, Reiner %T Closed surfaces with bounds on their Willmore energy %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2012 %P 605-634 %V 11 %N 3 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2012_5_11_3_605_0/ %G en %F ASNSP_2012_5_11_3_605_0
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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