Some functional properties on Cartan-Hadamard manifolds of very negative curvature

2021/05/19 by Ludovico Marini, Marini, Ludovico, Giona Veronelli +1
Mathematics · #35A23 #35J10 #46E35 #53C21 #58J05 #58J65 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2105.09024

openalex publication_date 2021/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider Cartan-Hadamard manifolds (i.e. simply connected of non-positive sectional curvature) whose negative Ricci curvature grows polynomially at infinity. We show that a number of functional properties, which typically hold when the curvature is bounded, remain true in this setting. These include the characterization of Sobolev spaces on manifolds, the so-called Caldéron-Zygmund inequalities and the L^p-positivity preserving property, i.e. u∈ L^p & (-Δ+ 1)u≥ 0 ⇒ u≥ 0. The main tool is a new class of first and second order Hardy-type inequalities on Cartan-Hadamard manifolds with a polynomial upper bound on the curvature. In the last part of the manuscript we prove the L^p-positivity preserving property, p∈[1,+∞], on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. Güneysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.

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