Date & Time: Friday, March 08, 2013 at 4 p.m.
Venue: Ramanujan Hall
Title:Navigating the Space of Surfaces
Speaker: Prof. Ulrich Pinkall, Technical University, Berlin
Abstract: Those functions that arise as the curvature function k of some closed plane curve of length L form a submanifold M of codimension three in the vector space of all L-periodic functions. M can be considered as the "shape space" for plane curves. Here we provide a similar approach for surfaces in space. For example, by the uniformization theorem, every smooth topological sphere in space admits a conformal parametrization f: S^2 ---> R^3. The role previously played by k now is taken by the so-called "mean curvature half density" H|df| induced on S^2 via f. Those half-densities on S^2 that arise as mean curvature half-densities of surfaces now form a hypersurface M in the euclidean vector space of all half-densities. Working on M instead of dealing directly with the immersions f dramatically simplifies many operations on surfaces that previously either were not tractable or required hard analysis. One striking example is provided by the gradient flow of the Willmore functional int (H^2). Other applications arise in Computer Graphics where it sometimes is more useful to modify the curvature of a surface than to try to handle the point positions directly."