Veranstaltungsübersicht -- Sommersemester 2012 (Archiv)

We present a numerical method for the computation of n-dimensional Ricci-DeTurck flow. Ricci flow is a geometric flow deforming a time-dependent metric on a Riemannian manifold proportional to its Ricci curvature. Ricci-DeTurck flow is a reparametrization of this flow using the harmonic map flow in order to get a strictly parabolic PDE. Our numerical method is based on the assumption that the manifold is orientable and differentiably embeddable in \R^{n+1}. By this means, it is possible to do computations in the basis of the ambient space. A weak formulation of Ricci-DeTurck flow is derived such that it only contains tangential gradients. A spatial discretization of this formulation with finite elements on polyhedral hypersurfaces and an implicit time discretization lead to an algorithm for computing Ricci-DeTurck flow. We have performed numerical tests for two and three dimensional hypersurfaces using piecewise linear finite elements. The generalization to non-orientable hypersurfaces of higher codimensions is still open.

We consider a branched Willmore surface immersed in $\R^{m\ge3}$ with square-integrable second fundamental form. We develop around each branch point local asymptotics for the immersion, its first, and its second derivatives. These expansions are given in terms of a first residue" (constant vector in $\R^m$) and in terms ofsecond residues" (integer-valued). These residues are computed as circulation integrals around each branch point. We then deduce explicit point removability conditions in terms of the residues, ensuring that the (branched) immersion is smooth across its branch points. Do residues pass through the weak limit? We'll see... [Talk based on a joint-work with Tristan Rivière]