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

Abstract

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 follows from a new generalized compactness theorem, which identifies the limit of a sequence of sets $E_i\subset X_i$ with uniformly bounded measure and perimeter, where $(X_i,{\mathsf{d}}_i,{\mathcal{H}}^N)$ is an arbitrary sequence of $\mathsf{RCD}(K,N)$ spaces. An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.

arXiv admin note: text overlap with arXiv:2107.07318

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• Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds
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Cited by in corpus (3)

• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
• From bubbles to clusters: Multiple solutions to the Allen--Cahn system