Stationary solutions to the Keller-Segel equation on curved planes

TL;DR

This paper investigates stationary solutions to the Keller-Segel equation on curved surfaces, establishing conditions for existence, nonexistence, and bounds related to the total mass, with novel results extending to non-flat geometries.

Contribution

It introduces a new correspondence between stationary solutions on curved planes and positively curved metrics, and proves a curved Hardy-Littlewood-Sobolev inequality, extending classical results.

Findings

Mass must be 8π for stationary solutions.

Existence of metrics close to flat that lack solutions.

Keller-Segel free energy bounded below only at mass 8π.

Abstract

We study stationary solutions to the Keller--Segel equation on curved planes. We prove the necessity of the mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require the solutions to have a finite second moment, and thus are novel already in the flat case. Furthermore, we provide a correspondence between stationary solutions to the static Keller--Segel equation on curved planes and positively curved Riemannian metrics on the sphere. We use this duality to show the nonexistence of solutions in certain situations. In particular, we show the existence of metrics, arbitrarily close to the flat one on the plane, that do not support stationary solutions to the static Keller--Segel equation (with any mass). Finally, as a complementary result, we prove a curved version of the logarithmic Hardy--Littlewood--Sobolev inequality and use it to show that the Keller--Segel free energy is bounded from below exactly when the mass is $8 \pi$, even in the curved case.