Curvature bounds and heat kernels

Abstract

<p>We introduce and study rough (approximate) lower curvature bounds and rough curvature-dimension conditions for discrete spaces and for graphs. These notions extend the ones introduced in \cite{St06a} and \cite{St06b} to a larger class of non-geodesic metric measure spaces. They are stable under an appropriate notion of convergence in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature $\geq K$ will have curvature $\geq K$ in the sense of \cite{St06a}. Moreover, in the converse direction, discretizations of metric measure spaces with curvature $\geq K$ will have rough curvature $\geq K$. We apply our results to concrete examples of homogeneous planar graphs. We derive perturbed transportation cost inequalities, that imply mass concentration and exponential integrability of Lipschitz maps. For spaces that satisfy a rough curvature-dimension condition we prove a generalized Brunn-Minkowski inequality and a Bonnet-Myers type theorem. <br /> Furthermore, we study Dirichlet forms on finite graphs and their approximations by Dirichlet forms on tubular neighborhoods. Our approach is based on a functional analytic concept of convergence of operators and quadratic forms with changing $L_2$-spaces, which uses the notion of measured Gromov-Hausdorff convergence for the underlying spaces. The convergence of the Dirichlet forms entails the convergence of the associated semigroups, resolvents and spectra to the corresponding objects on the graph.</p>