A good model describing droplets at thermodynamical equilibrium consists in volume constrained solutions of the surface energy combined with a potential gradient arising from an external potential field. With the main goal of approaching this phenomena in curved ambient spaces, in a work joint with Shrey Aryan (MIT) we prove a first classification of minimizers for a rich family of radial weights. We also provide counter examples to show that, in contrast with the weighted isoperimetric case, stability of centered spheres in general does not imply global minimiality. Time permitting, we will discuss a quantitative sharp stability result based on the previous classification.