Calculus of Variations and Geometric Measure Theory
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G. Antonelli - E. Pasqualetto - M. Pozzetta - D. Semola

Sharp isoperimetric comparison and asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

created by antonelli on 14 Jan 2022



Inserted: 14 jan 2022

Year: 2022

ArXiv: 2201.04916 PDF


This paper studies sharp and rigid isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called $N$-dimensional $\mathrm{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Thanks to these results, we determine the asymptotic isoperimetric behaviour for small volumes in great generality, and for large volumes when $K=0$ under an additional noncollapsing assumption. Moreover, we obtain new stability results for isoperimetric regions along sequences of spaces with uniform lower Ricci curvature and lower volume bounds, almost regularity theorems formulated in terms of the isoperimetric profile, and enhanced consequences at the level of several functional inequalities.\\ The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences.\\ Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies.

Keywords: isoperimetric inequality, isoperimetric problem, RCD space, isoperimetric profile, Lower Ricci bounds

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