Manuel Ritoré

Proof of the Double Bubble Conjecture

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3 .

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍn

AbstractIn this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍnwhich are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍn.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 ℍn.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 ℍn.

Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

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.

Stable constant mean curvature tori and the isoperimetric problem in three space forms

Introduction Let ψ : M → N be an immersion of an orientable surface into a three dimensional oriented Riemannian manifold. Then ψ has constant mean curvature if and only if it is a critical point of the area functional for any compactly supported variation that preserves the volume enclosed by the surface. In this context we say that the constant mean curvature immersion ψ is stable if the second variation formula of the area, which we call henceforth the index form of ψ, is non negative for all variations of the above type. Otherwise, ψ is stable if for any f ∈ C∞(M) with compact support such that ∫ M f dA = 0, we have

Isoperimetric regions in cones

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.

The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds

It is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in (non fixed) flat three manifolds are compact in a strong sense: given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the Ck topology. These limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds. Some applications to the isoperimetric problem are given.

Existence of isoperimetric regions in contact sub-Riemannian manifolds

Abstract We prove existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds whose quotient by the group of contact transformations preserving the sub-Riemannian metric is compact.

Proof of the double bubble conjecture

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3 .

Mean curvature flow and isoperimetric inequalities

The classical isoperimetric inequality in Euclidean space. Three different approaches.- The curve shortening flow and isoperimetric inequalities on surfaces.-

The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds

H^k