Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension

TL;DR

This paper explores isoperimetric and concentration properties of weighted Riemannian manifolds with negative generalized dimension, identifying new model spaces and establishing inequalities under various curvature conditions.

Contribution

It extends the Curvature-Dimension condition framework to negative dimensions, discovering new model spaces and deriving inequalities for these cases.

Findings

Identification of a new model space: a positively curved sphere with negative dimension.

Weak concentration implies N-dimensional Cheeger isoperimetric inequality.

Spaces with positive curvature satisfy a uniform Poincaré inequality and exhibit two-level concentration.

Abstract

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \in [n,\infty]$. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound, and discover a new case yielding a \emph{single} model space (besides the previously known $N$-sphere and Gaussian measure when $N \in [n,\infty]$): a (positively curved) sphere of (possibly negative) dimension $N \in (-\infty,1)$. When curvature is non-negative, we show that arbitrarily weak concentration implies an $N$-dimensional Cheeger isoperimetric inequality, and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincar\'e inequality uniformly for all $N \in (-\infty,1-\epsilon]$, and enjoy a two-level concentration of the type $\exp(-\min(t,t^2))$. Our main technical tool is a generalized version of the Heintze--Karcher theorem, which we extend to the range $N \in (-\infty,1)$.