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Pure Maths Colloquium: A proof of the Willmore Conjecture
Presented by "Martin Schmidt, FU Berlin, Germany",
20 May 2002 00:00 in CM221
" The global Weierstrass representation is used to transform the variational problem of the Willmore functional on tori
immersed in R3 into a variational problem with constraints on the space of Fermi curves of two--dimensional Dirac
operators with periodic potentials. These Dirac operators are the Lax operators of an integrable system. The Fermi
curves are the integrals of motion. The Willmore functional is the corresponding first integral. In turns out that the
moduli space of these Fermi curves is not complete. After adding the Fermi curves of finite rank perturbations of such
Dirac operators the subsets of the moduli space, on which the first integral is bounded, becomes compact. We show
that the Fermi curves of relative minimizers of this contrained variational problem have dividing real parts. This allows
to classify all relative minimizers, whose first intgeral is not larger than 8\pi. The total minimum is realized by the Fermi
curve of the Clifford torus. "
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