On stability of capillary surfaces in a ball

Abstract

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 . Consider a smooth and compact convex body B in R 3 . Let @B and intB denote its boundary and its interior respectively. We are interested in em- bedded constant mean curvature surfacesM inR 3 with non empty boundary such that intM intB and @M@B and which intersect @B at a constant angle 2 (0;). Such surfaces, called capillary surfaces, are critical points of an energy functional under some constraints. The energy functional is de- ned as follows: the surface M separates B into two bodies, consider among these two bodies the one inside which the angle is measured and call the part of its boundary that lies on @B. Denote by A the area of M and by T that of . The energy function is then E = A cosT: The space of surfaces under consideration are compact orientable surfaces in R 3 with boundary contained in @B and interior contained in intB and which divide B into two bodies of preassigned volumes. The Euler-Lagrange equation shows that a critical point ofE under these constraints is a constant mean curvature surface that intersect @B at the constant angle , that is the angle between the exterior conormals to @M in M and is everywhere equal to along @M. We say that such a surface is capillarily stable if it minimizes the energy up to second order. Capillary surfaces correspond to the physical problem of the behavior of an incompressible liquid in a container B in the absence of gravity. A great deal of work has been devoted to capillary phenomena from the point of view of existence and uniqueness of solutions mainly in the non-parametric case and in the more general situation of presence of gravity (see the book of R. Finn, (F), for an account of the