2022/04/15 by Mohammad Ghomi, Joel Spruck, Ghomi, Mohammad +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #49Q15 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Myofascial pain diagnosis and treatment #Point processes and geometric inequalities #Primary: 53C20 #Secondary: 52A38
paper · pdf · doi:10.48550/arxiv.2204.07624
openalex publication_date 2022/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive several geometric inequalities for a convex hypersurface Γ in a Cartan-Hadamard manifold M. In particular we show that the first mean curvature integral of a convex hypersurface γ nested inside Γ cannot exceed that of Γ, which leads to a sharp lower bound in dimension 3 for the total first mean curvature of Γ in terms of the volume it bounds in M. This monotonicity property is extended to all mean curvature integrals when γ is parallel to Γ, or M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.