We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3 . * 2000 Mathematics Subject Classification. Primary: 53A10. Secondary: 53C42. ing curves no longer meet at 120 degrees.) Unfortunately, for general costs, even for equal volumes, (7.1) does not imply both regions connected. We thank undergraduate Ken Dennison for raising this question.

We prove that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space. We classify least area surfaces in the quotient of R 3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact twosided index one minimal surfaces in non-negatively curved ambient spaces. Finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface.

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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 .

Abstract. The partitioning problem for a smooth convex body B C IR 3 consists in to study, among surfaces which divide B in two pieces of prescribed volume, those which are critical points of the area functional.We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case that B is a Euclidean ball we obtain stronger results.Mathematics Subject Classifications (1991): 53A10, 49Q05, 49Q10.