Abstract:

We study the volume-preserving flow of smooth, closed, and convex hypersurfaces in hyperbolic space with the speed given by arbitrary positive power of the Gauss curvature. We prove that if the initial hypersurface is convex, the solution remains convex and exists for all positive times. Furthermore, by applying a result of Kohlmann—which characterizes geodesic balls in terms of hyperbolic curvature measures—we show that the flow converges smoothly to a geodesic sphere. This presents the first result for (globally constrained)  volume-preserving curvature flows in hyperbolic space that only requires initial convexity. This is joint work with Bo Yang (Tsinghua University) and Tailong Zhou (Sichuan University).