Abstract:

Pogorelov's rigidity theorem states that a compact convex body in the hyperbolic 3-space is determined up to isometry by the intrinsic path metric on its boundary.  In this talk, we show that the intrinsic path metric on the boundary determines a closed non-compact convex set up to isometry, provided that the set of limit points of the convex set at infinity of the hyperbolic 3-space has vanishing 1-dimensional Hausdorff measure, i.e.,  zero length.  Furthermore, this zero-length condition is optimal.  This can be considered as an analogue of the Painleve removable singularity theorem in complex analysis, which states that compact sets of zero length are removable for bounded holomorphic functions.    This is a joint work with Yanwen Luo and Zhenghao Rao.

Bio:

Feng Luo is a professor of Math at Rutgers University.  He obtained his BS from Peking University and Ph.D from UC San Diego.