Smooth approximation, Alexandrov-Fenchel type inequalities and curvature flows for h-convex bodies in hyperbolic space

Abstract

We associate each h-convex body in hyperbolic space $\mathbb{H}^{n+1}(n\geq 2)$ with a Radon measure valued $(0,2)$-tensor on the sphere $\mathbb{S}^n$ through its horospherical function. Using spherical convolution, we prove that any h-convex body can be approximated by a sequence of smooth strictly h-convex bodies in the Hausdorff sense. This approximation result extends the inequality parts of the Alexandrov-Fenchel inequality and the modified Alexandrov-Fenchel inequality to general h-convex bodies, but equality rigidity cannot be obtained directly. We prove that the equality holds in the Alexandrov-Fenchel inequality and the modified Alexandrov-Fenchel inequality if and only if the h-convex bodies are geodesic balls. For the Alexandrov-Fenchel inequality, we establish the Hausdorff stability independent of curvature bounds described by the Alexandrov-Fenchel deficits for h-convex bodies, which implies the equality rigidity of the Alexandrov-Fenchel inequality. For the modified Alexandrov-Fenchel inequality, we introduce a class of globally constrained $k$-th modified quermassintegrals preserving flows with general h-convex initial hypersurfaces to obtain the equality rigidity. Moreover, we prove the flows above preserve h-convexity, make the $(k+1)$-th modified quermassintegrals non-increasing and converge to geodesic spheres. We prove a rigidity theorem for modified area measures for h-convex bodies and use this rigidity theorem to characterize the geodesic balls via modified support measures for h-convex bodies, from which we prove that the $(k+1)$-th modified quermassintegrals strictly decrease in a short time along the flows with non-spherical initial hypersurfaces. Then we obtain the equality rigidity of the modified Alexandrov-Fenchel inequality.