Has WIllmore Conjecture completely been solved?
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From: Pioneer on 29 Sep 2009 23:45 Has WIllmore Conjecture completely been solved? A proof of the Willmore conjecture Authors: Martin Ulrich Schmidt (Submitted on 21 Mar 2002 (v1), last revised 22 Mar 2002 (this version, v2)) Abstract: A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all spectral curves corresponding to periodic solutions of the Davey-Stewartson equation. The subsets of this moduli space, which correspond to bounded first integrals, are shown to be compact. With respect to another topology the moduli space is shown to be a Banach manifold. The subset of all periodic solutions of the Davey-Stewartson equation, which correspond to immersion of tori into the three-dimensional Euclidean space, are characterized by a singularity condition on the corresponding spectral curves. This yields a proof of the existence of minimizers for all conformal classes and the determination of the absolute minimum, which is realized by the Clifford torus. Comments: 215 pages Subjects: Differential Geometry (math.DG); Mathematical Physics (math- ph); Analysis of PDEs (math.AP); Spectral Theory (math.SP) MSC classes: 37K25;32G13 Cite as: arXiv:math/0203224v2 [math.DG] A proof of the Willmore conjecture Martin U. Schmidt* Institut f¨ur Theoretische Physik Freie Universit¨at Berlin Arnimallee 14 D-14195 Berlin |