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

TL;DR

This paper investigates spectral properties and functional inequalities on manifolds with conical singularities, where traditional curvature conditions do not apply, and establishes new inequalities and spectral gap estimates in this setting.

Contribution

It introduces a version of Bakry-Émery inequality, a novel Hardy inequality, and spectral gap estimates for manifolds with conical singularities, extending analysis beyond classical curvature bounds.

Findings

Established a Bakry-Émery inequality for singular manifolds.

Proved a new Hardy inequality applicable to conical singularities.

Derived spectral gap estimates in the presence of conical singularities.

Abstract

We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension $\ge3$ 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-\'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.