On stability of capillary surfaces in a ball

We study stable capillary surfaces in a euclidean ball in the absence of gravity. We prove, in particular, that such a surface must be a flat disk or a spherical cap if it has genus zero. We also prove that its genus is at most one and it has at most three connected boundary components in case it is minimal. Some of our results also hold in H 3 and S 3. Introduction. Consider a smooth and compact convex body B in R 3. Let ∂B and int B denote its boundary and its interior respectively. We are interested in embedded constant mean curvature surfaces M in R 3 with non empty boundary such that int M ⊂ int B and ∂M ⊂ ∂B and which intersect ∂B at a constant