TAILIEUCHUNG - Đề tài " Removability of point singularities of Willmore surfaces "

We investigate point singularities of Willmore surfaces, which for example appear as blowups of the Willmore flow near singularities, and prove that closed Willmore surfaces with one unit density point singularity are smooth in codimension one. As applications we get in codimension one that the Willmore flow of spheres with energy less than 8π exists for all time and converges to a round sphere and further that the set of Willmore tori with energy less than 8π − δ is compact up to M¨bius transformations. o 1. Introduction For an immersed closed surface f : . | Annals of Mathematics Removability of point singularities of Willmore surfaces By Ernst Kuwert and Reiner Sch atzle Annals of Mathematics 160 2004 315 357 Removability of point singularities of Willmore surfaces By Ernst Kuwert and Reiner Schatzle Abstract We investigate point singularities of Willmore surfaces which for example appear as blowups of the Willmore flow near singularities and prove that closed Willmore surfaces with one unit density point singularity are smooth in codimension one. As applications we get in codimension one that the Willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of Willmore tori with energy less than 8 Ỗ is compact up to Mobius transformations. 1. Introduction For an immersed closed surface f s Rra the Willmore functional is defined by W f 4 H 2 . where H denotes the mean curvature vector of f g f geuc the pull-back metric and IX the induced area measure on s. The Gauss equations and the Gauss-Bonnet Theorem give rise to equivalent expressions W f 4 i A 2 d g X s 1 i A 2 d g 2 X s 4 J 2 J s s where A denotes the second fundamental form A A 1 g H its trace-free part and X the Euler characteristic. The Willmore functional is scale invariant and moreover invariant under the full Mobius group of Rra. Critical points of W are called Willmore surfaces or more precisely Willmore immersions. We always have W f 4 with equality only for round spheres see Wil in codimension one that is n 3. On the other hand if W f 8 E. Kuwert was supported by DFG Forschergruppe 469. R. Schatzle was supported by DFG Sonderforschungsbereich 611 and by the European Community s Human Potential Programme under contract HPRN-CT-2002-00274 fRoNTS-SINGULARITIES. 316 ERNST KUWERT AND REINER SCHATZLE then f is an embedding by an inequality of Li and Yau in LY for the reader s convenience see also in our appendix. Bryant classified in Bry all Willmore spheres in codimension one. In KuSch 2 .

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