Geometry and Topology Seminar[ Edit ]
Moderator: Michael Khanevsky
In this talk, the speaker aims to present his current investigation of the ``Moebius-invariant Willmore flow'', which is a conformally invariant modification of the classical Willmore-flow, i.e. of the gradient flow of the Willmore functional. Since the Willmore functional - applied to immersions f of some fixed closed surface into Euclidean space - can be defined as the integral over the squared modulus of the trace-free part of the second fundamental form of f plus 2 pi times the Euler-characteristic of the underlying surface, it can be geometrically interpreted as a global conformal invariant which measures the deviation of immersions from being totally umbilic. In 2001--2004, Kuwert and Schaetzle proved in 3 consecutive articles that the classical Willmore flow moving immersions of some fixed 2-sphere into Euclidean space exists globally and actually converges smoothly to a round sphere, if the initial Willmore energy is smaller than 8 pi. Only this year, the speaker invented a new ``descent-technique'' in order to prove that the Moebius-invariant Willmore flow moving immersions of a fixed torus into the 3-sphere exists globally if the start immersion maps this torus onto a Hopf-torus in the 3-sphere, and that it actually converges smoothly to some conformal image of the Clifford-torus in the 3-sphere, if the initial Willmore energy is smaller than a certain number between 8 pi and 9 pi. In order to precisely compute this concrete threshold, the speaker reduced the Euler-Lagrange equation of the Willmore functional for Hopf-tori to a non-linear ODE of first order which can be integrated in terms of elliptic functions - an adaption of Einstein's computation of the annual shift of the perihelion of a relativistic planetary orbit - and then continued to apply classical methods and formulas of Arithmetic Geometry and Analytic Number Theory.
An aperiodic tiling of R^d has a hull, which typically is a compact foliated space X. The hull comes equipped with an invariant transverse measure, corresponding to a Z^d-invariant measure on a totally disconnected transversal N. Mathematical physicists attach two types of invariants to this situation. On the one hand, the invariant transverse measure gives rise to a real-valued map on H^d(X; R) (Cech cohomology) which corresponds to the diffraction pattern aspect of tilings. On the other hand. it also gives rise to a real-valued map on K_0(C^*G(X)), the topological K-theory of the C^*-algebra of the holonomy groupoid of the tiling, which corresponds to the spectral (of Hamiltonians defined on such tilings) aspect of tilings. We use the Index Theorem for foliated spaces to prove that under some general assumptions that these points of view are equivalent.
This is joint work with Eric Akkermans (physics, the Technion) and Jonathan Rosenberg (math, U. Maryland).