We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension \(\ge 3\) with singularities. Of particular interest will be manifolds 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 1 \(\begin{aligned} k(x)\simeq K_i-\frac{s_i}{d^2(z_i,x)}. \end{aligned}\) 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 in the sense of Erbar et al. (J Math Pures Appl 161: 1–69, 2022). Manifolds with such a singular Ricci bound (1) appear quite naturally. The prime examples are metric cones, for instance, \(M={{\mathbb {R}}}_+\times _r N\) with any \((N,g^N)\) satisfying \(\inf _{y\in N}\textrm{Ric}_y^N<(n-2)g^N\) , e.g. spheres \(N={{\mathbb {S}}}^{n-1}_R\) with radius \(R>1\) .
For manifolds with such conical singularities we will prove a version of the Bakry–Émery inequality
a novel Hardy inequality
a spectral gap estimate.
Related examples are provided by weighted spaces, e.g. \(M={{\mathbb {R}}}^n\) with \(g=g^{Euclid}\) and \(m(dx)=|x|^\alpha d{\mathfrak {L}}^n(x)\) for some \(\alpha \in {{\mathbb {R}}}\) where the largest lower bound for Bakry–Émery Ricci tensor is given by \( k(x)=-\frac{|\alpha |}{|x|^2}\) , and
Grushin-type spaces \(M={{\mathbb {R}}}^j \times _f {{\mathbb {R}}}^{n-j}\) with \(f(y)=|y|^{-\alpha }\) for suitable \(\alpha >0\) , either with Riemannian volume measure or with Lebesgue measure, which admit lower Ricci bounds of the form \(k(y,z)=-\frac{C}{|y|^2}\) .
