Stationary solutions to the Keller–Segel equation on curved planes

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