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

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 $N \in (-\infty,1)$.