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

2024/05/17 by Sturm, Karl-Theodor · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2405.10734

Abstract

We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension ≥3 with (finite or countably many) conical singularities \z_i\_{i∈\mathfrak I} in the neighborhood of which the largest lower bound for the Ricci curvature is k(x)≃ K_i-(s_i)/(d²(z_i,x)). 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.

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• Comparison estimates on nonsmooth spaces with integrable Ricci lower bounds via localization

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