Event № 943
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.