Bakry-Èmery, Hardy, and Spectral Gap Estimates on Manifolds with Conical Singularities

Abstract

We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension $\ge 3$ with (finite or countably many) conical singularities $\{z_i{\}}_{i\in \mathfrak{I}}$ in the neighborhood of which the largest lower bound for the Ricci curvature is \begin{equation}\label{d2} k(x)\simeq K_i-\frac{s_i}{d^2(z_i,x)}. \end{equation} Thus none of the existing Bakry-Émery inequalities or curvature-dimension conditions apply. In particular, $k$ does not belong to the Kato (or (extended Kato) class, and $(M,g)$ is not tamed. Manifolds with such a singular Ricci bound appear quite naturally., e.g. as cones over spheres of radius $>1$ For such manifolds with conical singularities we will prove * a version of the Bakry-Émery inequality * a novel Hardy inequality * a spectral gap estimate.