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 Poincaré 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 $N\in (-\infty ,1)$.

arXiv comment38 pages. Corrected Typo in the formulation of Theorem 1.3 (2). Final version to appear in Trans. of Amer. Math. Soc