Continuity and differentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

Abstract

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. Then under an extra hypothesis on the geometry of $M$, we apply this result to prove some differentiability property of $I_M$ and a differential inequality satisfied by $I_M$, extending in this way well known results for compact manifolds, to this class of noncompact complete Riemannian manifolds with bounded geometry.

31 pages, 8 figures. A new entire section, namely section $2$ is added to give the details of the equivalence between the weak and strong formulation of the isoperimetric problem, some typos are corrected

References in corpus (4)

• Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions
• The isoperimetric problem of a complete Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends
• A discontinuous isoperimetric profile for a complete Riemannian manifold
• On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

Cited by in corpus (7)

• Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions
• Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian $3$-manifolds
• The isoperimetric problem of a complete Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends
• Sharp isoperimetric inequalities for small volumes in complete noncompact Riemannian manifolds of bounded geometry involving the scalar curvature
• On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry
• Rigidity of Hawking mass for surfaces in three manifolds
• A discontinuous isoperimetric profile for a complete Riemannian manifold