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

TL;DR

This paper investigates functional properties on Cartan-Hadamard manifolds with very negative curvature, demonstrating that key inequalities and space characterizations hold under polynomial Ricci curvature growth, extending known results to broader geometric settings.

Contribution

It introduces new Hardy-type inequalities and proves the $L^p$-positivity preserving property for manifolds with polynomial Ricci curvature growth, generalizing previous results.

Findings

Sobolev space characterizations extend to these manifolds.

Caldón-Zygmund inequalities remain valid.

The $L^p$-positivity preserving property holds under polynomial Ricci curvature growth.

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 Cald\'eron-Zygmund inequalities and the $L^p$-positivity preserving property, i.e. $u\in L^p\ \&\ (-\Delta + 1)u\ge 0 \Rightarrow u\ge 0$. 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, $p\in[1,+\infty]$, on manifolds with subquadratic negative part of the Ricci curvature. This generalizes an idea of B. G\"uneysu and gives a new proof of a well-known condition for the stochastic completeness due to P. Hsu.