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.

Dedicated to Professor Manfredo do Carmo on the occasion of his 80 th birthdayOne of the most celebrated papers by Manfredo is his joint work with Peng [19] on the classification of complete orientable stable minimal surfaces in R 3 . The authors, through a clever use of the second variation formula of the area together with some information on the conformal geometry of the surface, provided a beautiful and straightforward proof of the following

Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists uε, a critical point of the Allen-Cahn energy Eε(u) = ε 2 |∇u| 2 + (1 − u 2 ) 2 , whose nodal set converges to Σ as ε tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function uε being a critical point of Eε under some volume constraint.

Abstract. We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point.We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.

In this paper we study sets in the n-dimensional Heisenberg group H n which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in H n . We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones.Our main result describes which are the CMC hypersurfaces of revolution in H n . The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence we classify the rotationally invariant isoperimetric sets in H n .

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Abstract. We consider cones C = 0 × × M n and prove that if the Ricci curvature of C is nonnegative, then geodesic balls about the vertex minimize perimeter for given volume. If strict inequality holds, then they are the only stable regions.