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Title: Total mean curvatures of Riemannian hypersurfaces
Abstract 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 \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M 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.  more » « less
Award ID(s):
2202337
PAR ID:
10394464
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Advanced Nonlinear Studies
Volume:
23
Issue:
1
ISSN:
2169-0375
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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