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

Source abstract

We study the isoperimetric, functional and concentration properties of n n -dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension N N is negative and, more generally, is in the range N ∈ ( − ∞ , 1 ) N \in (-\infty ,1) , extending the scope from the traditional range N ∈ [ n , ∞ ] 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 single model space (besides the previously known N N -sphere and Gaussian measure when N ∈ [ n , ∞ ] N \in [n,\infty ] ): a (positively curved) sphere of (possibly negative) dimension N ∈ ( − ∞ , 1 ) N \in (-\infty ,1) . When curvature is non-negative, we show that arbitrarily weak concentration implies an N 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 ∈ ( − ∞ , 1 − ε ] N \in (-\infty ,1-\varepsilon ] and enjoy a two-level concentration of the type exp ⁡ ( − min ( t , t 2 ) ) \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 ∈ ( − ∞ , 1 ) N \in (-\infty ,1) .