Bakry-\'Emery, Hardy, and Spectral Gap Estimates on Manifolds with Conical Singularities

We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension ≥3 with (finite or countably many) conical singularities ᵢ∈ I in the neighborhood of which the largest lower bound for the Ricci curvature is {equation}{d2} k(x) K_i-s_i/d^2(z_i,x). {equation} Thus none of the existing Bakry-\'Emery 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-\'Emery inequality * a novel Hardy inequality * a spectral gap estimate.