The role of the curvature of a surface in the shape of the solutions to elliptic equations

We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-\mathrm{\Delta }u=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal.