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 \zᵢ\₈ ₈ in the neighborhood of which the largest lower bound for the Ricci curvature is equationd2 k (x) Kᵢ-sᵢd² (zᵢ, 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.