Submanifold

Inequality

Engineering

Curvature

Pure mathematics

Boundary (topology)

Ricci curvature

Riemannian manifold

Limit (mathematics)

Geology

Oceanography

Mechanical engineering

Obstacle

Mathematics

Hull

Political science

Mathematical analysis

Geometry

Combinatorics

Manifold (fluid mechanics)

Isoperimetric inequality

Journal of Functional Analysis

The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull Ω⁎ of a set Ω is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where Ω has C1,α-boundary, the area of ∂Ω⁎ is recovered as the limit of the p-capacities of Ω, as p→1+. Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided 3≤n≤7.

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds