Aggregation-diffusion energies on Cartan-Hadamard manifolds of unbounded curvature

TL;DR

This paper studies the existence of ground states for an aggregation-diffusion energy on Cartan-Hadamard manifolds with unbounded curvature, deriving explicit conditions on the attractive potential and establishing a new inequality.

Contribution

It provides necessary and sufficient conditions for ground state existence on manifolds with unbounded curvature and introduces a novel logarithmic Hardy-Littlewood inequality.

Findings

Derived explicit conditions on the attractive potential for ground states.

Established a new logarithmic Hardy-Littlewood inequality for unbounded curvature manifolds.

Connected curvature growth with the existence of equilibrium states.

Abstract

We consider an aggregation-diffusion energy on Cartan-Hadamard manifolds with sectional curvatures that can grow unbounded at infinity. The energy corresponds to a macroscopic aggregation model that involves nonlocal interactions and linear diffusion. We establish necessary and sufficient conditions on the growth at infinity of the attractive interaction potential for ground states to exist. Specifically, we derive explicit conditions on the attractive potential in terms of the bounds on the sectional curvatures at infinity. To prove our results we establish a new logarithmic Hardy-Littlewood inequality for Cartan-Hadamard manifolds of unbounded curvature.