Isoperimetric inequality in noncompact MCP spaces

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

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 compl…

Local Hölder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-Hölder continuous function and so in particular it 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. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own int…

The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space $X$ along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result f…

Generalized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profile

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to such noncompact manifolds.

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

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 $\mathscr{C}^{1, α}$-boundary, the area of $\partial Ω^*$ is recovered as the limit of the $p$-capacities of $Ω$, as $p \to 1^+$. Finally, building on the existence of strictly …

具有非负Ricci曲率的非紧流形中大等周集的平均唯一性

Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature

该研究在具有非负Ricci曲率、欧几里得体积增长和二次曲率衰减的非紧完备黎曼流形中，证明了大体积等周集在体积趋于无穷时的平均唯一性及边界严格保体积稳定性，并构造反例说明无法对所有大体积都成立唯一性或严格稳定性。