Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds

By A. V. Kolesnikov, Emanuel Milman. 2017

54 citations recorded by the source corpus.

abstract:We study a Riemannian manifold equipped with a density which satisfies the Bakry-\'Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincar\'e-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski inequality, thereby precluding any extensions beyond the Euclidean setting. A dual version for generalized mean-convex boundaries is also obtained, yielding spectral-gap estimates for the weighted Laplacian on the boundary. Motivated by these inequalities, a new geometric evolution equation is proposed, which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion thus far confined to the purely linear setting. This geometric flow is characterized by having parallel normals (of varying velocity) to the evolving hypersurface along the trajectory, and is intimately related to a homogeneous Monge-Amp\`ere equation on the exterior of the convex domain. Using the aforementioned Poincar\'e-type inequality on the boundary of the evolving hypersurface, we obtain a novel Brunn-Minkowski inequality in the weighted-Riemannian setting, amounting to a certain concavity property for the weighted-volume of the evolving enclosed domain. All of these results appear to be new even in the classical non-weighted Riemannian setting.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/d10c32142c8aca731123f725503af0a74e0f8818

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

The dimensional Brunn–Minkowski inequality in Gauss space (2020, background): “(28) In order to prove Proposition 8 one has to repeat the arguments of the proof of Proposition 2 in [KL18] (see also [KM18]) mutatis mutandis, by replacing the power function t 7→ t δ n by a general increasing function ψ.”
Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, background): “In fact, our proof in [36] applies to a general weighted Riemannian manifold satisfying the Curvature-Dimension condition CD(0, N), yielding a novel interpretation of Minkowski addition in the Riemannian setting.”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, result): “An extension of Milman’s results is due to Cavalletti and Mondino [13], who used the OMT-approach in combination to themeasure disintegrationmethod of Klartag [33] to prove a sharp Lévy-Gromov-type isoperimetric inequality on metric measure spaces satisfying the CD(κ, N ) condition with κ > 0 and N ”
Local Lp-Brunn-Minkowski inequalities for p<1 (2017, methodology): “he local p-BM inequality We now derive a suﬃcient condition for conﬁrming the local p-BM inequality (3.3) in its equivalent inﬁnitesimal form (4.16) on ∂K. Our motivation comes from our previous work [36], where we obtained a new proof of the (local, and hence global) Brunn–Minkowski inequality (p =”