  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.
