Xiaodan Zhou - Characterization of horizontal quasiconvexity in the Heisenberg group and applications

This talk is concerned with PDE-based characterizations of horizontal quasi-convexity in the Heisenberg group H. For upper semicontinuous, h-quasiconvex functions, we provide a characterization in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation and a sufficient condition in terms of a second-order PDE. Applications of these characterizations include constructing horizontally quasiconvex envelope of a continuous function, construct h-convex hull of a given set, and investigating the convexity preserving property of curvature flow in the Heisenberg group H. This talk is based on joint work with Antoni Kijowski, Qing Liu and Ye Zhang.