BEYOND TRADITIONAL CURVATURE-DIMENSION I: NEW MODEL SPACES FOR ISOPERIMETRIC AND CONCENTRATION INEQUALITIES IN NEGATIVE DIMENSION

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 epsilon (-infinity, 1), extending the scope from the traditional range N epsilon [n, infinity]. 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-sphere and Gaussian measure when N epsilon [n, infinity]): a (positively curved) sphere of (possibly negative) dimension N epsilon (-infinity, 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 Poincare inequality uniformly for all N epsilon (-infinity, 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 epsilon (-infinity, 1).