Antonio Ros

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 .

Properly embedded minimal surfaces with finite total curvature

Among properly embedded minimal surfaces in Euclidean three space, those that have finite total curvature from a natural and important subclass. The first nontrivial examples, other than the plane and the Catenoid, were constructed only some years ago by Costa [9], and Hoffman and Meeks [16], [17]. These examples began the study of existence, uniqueness and structure theorems for minimal surfaces of finite total curvature, usually attending to their topology. Several methods compete to solve the main problems in this theory, although up to now, the structure of the space of such kind of surfaces with a fixed topology is not well understood. However, we dispose today of a certain number of partial results, and some of them will be explained in these notes.

Stability for hypersurfaces of constant mean curvature with free boundary

The partitioning problem for a smooth convex bodyB ⊂ ℝ3 consists in to study, among surfaces which divideB 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 thatB is a Euclidean ball we obtain stronger results.

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

Some uniqueness and nonexistence theorems for embedded minimal surfaces

Thanks to the works of Callahan, Costa, Hoffman, Karcher, Meeks, Rosenberg, Wei, etc.—see for example [1],[2],[3],[5],[8],[11],[12],[16],[20]—, we dispose now of a large number of properly embedded minimal surfaces in the euclidean space IR other than the classical examples. All those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds. The most basic invariants associated to a surface of this type are its topology and its periodicity. It is an interesting problem to decide if the simplest examples—like the catenoid, the helicoid, the Scherk’s surfaces or the Riemann example—can be characterized in terms of some of these invariants. In the non-periodic case, there are two important uniqueness theorems in this direction: the first one, obtained by Schoen [17], characterizes the catenoid as the only complete minimal surface embedded

Embedded minimal surfaces: Forces, topology and symmetries

We prove topological uniqueness theorems for embedded minimal surfaces in ℝ3 under the assumption that certain forces associated to these surfaces are vertical. We give applications to minimal surfaces with symmetries and with free boundary.

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.

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 .

Stable constant-mean-curvature hypersurfaces are area minimizing in small L1 neighborhoods

Abstract.- We prove that a strictly stable constant-mean-curvature hypersurface ina smooth manifold of dimension less than or equal to 7 is uniquely homologically areaminimizing for fixed volume in a small L 1 neighborhood. 1. IntroductionBy work of White [W] and Grosse-Brauckman [Gr], a strictly stable constant-mean-curvature surface S 0 is minimizing in a small neighborhood U of S 0 amongcompetitor hypersurfaces S ⊂ U enclosing the same volume. Assuming M compact,we extend their results to a small L 1 neighborhood of S 0 , i.e., to hypersurfaces Ssuch that S − S 0 bounds a region with net volume 0 and small total volume.Stable constant-mean-curvature hypersurfaces in M appear in particular as so-lutions of the isoperimetric problem; see for instance [R1]. In the case that theambient space is a flat 3-torus T 3 there is a connection between the isoperimetricproblem and the study of mesoscale phase separation phenomena; see Choksi andSternberg [CS]. One simple model minimizes the Cahn-Hilliard free energyZ

The periodic isoperimetric problem

Given a discrete group G of isornetries of R 3 , we study the G-isoperimetric problem, which consists of minimizing area (modulo G) among surfaces in R 3 which enclose a G-invariant region with a prescribed volume fraction. If G is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where G = Pm3m (the group of symmetries of the integer rank three lattice Z 3 ) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than 1/6, and we give an isoperimetric inequality for G-invariant regions that, for instance, implies that the area (modulo Z 3 ) of a surface dividing the three space in two G-invariant regions with equal volume fractions, is at least 2.19 (the conjectured solution is the classical P Schwarz triply periodic minimal surface whose area is ∼ 2.34). Another consequence of this isoperimetric inequality is that Pm3m-symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group Z 3 .