Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds

• DOI: 10.48550/arxiv.math/0311304
• Publication Date: 2003-01-01
• Authors: Vincent Bayle; César Rosales
• Keywords: 53C20 (Primary) 49Q20 (Secondary); Differential Geometry (math.DG); FOS: Mathematics

We prove that the isoperimetric profile of a convex domain $Ω$ with compact closure in Riemannian manifold $(M^{n+1},g)$ satisfies second order differential inequality which only depends on dimension and lower bound Ricci curvature $Ω$. Regularity properties topological consequences regions arise naturally from this point view. Moreover, by integrating we obtain sharp comparison theorems: not can derive an should be compared Lévy-Gromov Inequality but also show if $\text{Ric}\geq nδ$ $Ω$, then is bounded above half-space $\mathbb{H}_δ^{n+1}$ simply connected space form constant sectional $δ$. As consequence comparisons geometric estimations for volume diameter first non-zero Neumann eigenvalue Laplace operator