Quantitative Isoperimetry à la Levy‐Gromov

Abstract

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the manifold and of the corresponding sphere is likewise bounded. These results are actually obtained in the more general context of (possibly nonsmooth) metric measure spaces with curvature‐dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method. © 2018 Wiley Periodicals, Inc.

Quantitative Obata's Theorem

“…A second crucial property of the decomposition {X q } q∈Q , inherited by the variational nature of the construction, is the so-called cyclical monotonicity. This was key in [CMM18] for showing that, for q ∈ Q ℓ , the transport ray X q has its starting point close to a fixed "south pole" P S , and ends-up nearby a fixed "north pole" P N (in particular, the distance between P S and P N is close to π) (Proposition 5.1).…”

“…"most rays are long"). As we will discuss in a few lines, this is far from being trivial (in particular, it needs new ideas when compared with [CMM18]).…”

“…The proofs of the two claims are the most technical part of the work and correspond respectively to Proposition 5.2 and Proposition 5.3. Let us mention that the two difficulties 1. and 2. were not present in the proof of the quantitative Lévy-Gromov inequality in [CMM18], where it was sufficient to work with characteristic functions (which have a fixed scale, i.e. they are either 0 or 1).…”

“…with diameter close to π we have a quantitative volume control for balls centred at extrema of long rays. The proof is inspired by [O07, Lemma 5.1] where the case of maximal diameter π is treated (see also [CMM18,Proposition 5.1]).…”

Stability of the Bakry-Émery theorem on Rn

“…Splitting theorems for manifolds satisfying a curvature bound and a geometric condition have been the topic of some interest, going back to work of Cheeger and Gromoll [18,17]. More recently, rigidity and stability for a related (and stronger) isoperimetric inequality has been established [15] under the stronger curvature-dimension condition with finite dimension, using completely different techniques.…”