Soap films bounded by non-closed curves

This paper proves (i) every “geometrically knotted” non-closed curve bounds a soap-film, (ii) any non-closed curve bounding a soap-film must have total curvature greater than 2π, and (iii) for every k > 2π, there is a geometrically knotted non-closed curve with total curvature k.     

Soap films and Kelvin's curved, truncated octahedron

Here, the Weierstrass representation theorem for minimal surfaces is used to derive parametrizations of two soap films spanning a rectangular prism with square base. The parametrizations are explicit, although the exact values of the height of the prism and the edge length of the square base can only be determined numerically from the formulas. The more famous of these two soap films can be extended to a partition of ℝ³ by curved, truncated octahedra. Furthermore, if the height of the prism is taken to be , then the partition is that discovered by Lord Kelvin in the late 19th century.     

Codimension one minimal cycles with coefficients inZ orZ p , and variational functionals on fibered spaces

Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H_* (M, Z) (or, respectively, in H_* (M, Z_p) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem. We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε. As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly the minimal cycle desired.     

In polytopes,small balls about some vertex minimize perimeter

In (the surface of) a convex polytope Pⁿ in ℝⁿ⁺¹, for small prescribed volume, geodesic balls about some vertex minimize perimeter.     

Minimal cones on hypercubes

It is shown that in dimension greater than four, the minimal area hypersurface separating the faces of a hypercube is the cone over the edges of the hypercube. This constrasts with the cases of two and three dimensions, where the cone is not minimal. For example, a soap film on a cubical frame has a small rounded square in the center. In dimensions over 6, the cone is minimal even if the area separating opposite faces is given zero weight. The proof uses the maximal flow problem that is dual to the minimal surface problem.     

Covering radii and paving diameters of Alexandrov spaces

Covering radii and paving diameters are defined, and the borderline case when cov_k X = π/2, k = 1,…,n + 1 and pav_k X = π/2, k = 1,…,n + 1 is studied (curv X ≥1, dim X = n).     

On C(K) Grothendieck spaces

Using the theory of bounded linear operators, several characterizations of C(K) Grothendieck spaces are given. Supported by CDCH of ULA under project 1123-02-05-B 2000.     

Interpolating varieties for spaces of meromorphic functions

We consider in this paper interpolation problems for weighted spaces of entire and meromorphic functions. Various (analytic and geometric) conditions necessary and sufficient for multiplicity varieties to be interpolating varieties will be given. This research is supported in part by NSF Grants DMS-90-00616 and CDR 88-03012.     

Besicovitch Covering Lemma,Hadamard manifolds,and zero entropy

It is proved that if the Besicovitch Covering Lemma is true on either a Hadamard manifold or a simply connected surface without focal points that covers a compact quotient, then the manifold is the Euclidean space. As a corollary, the vanishing of the topological entropy of a compact manifold of nonpositive curvature or of a compact surface without focal points is equivalent to the validity of the Besicovitch Covering Lemma on the universal covering space of the manifold. The author was partially supported by an NSF grant.     

Small excess and Ricci curvature

It is proved that a Riemanniann-manifold with Ricci curvature ≥ (n − 1) and a lower injectivity radius bound is a sphere provided the diameter is sufficiently close to π. The author was partially supported by the NSF and the Alfred P. Sloan Foundation.     

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-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 .     

The isoperimetric problem in complete surfaces of nonnegative curvature

We show that in a complete plane with nonnegative curvature there is a perimeter minimizing set of any given area. This set is a disc whose boundary is a closed embedded curve with constant geodesic curvature.     

Area minimizing sets subject to a volume constraint in a convex set

For a given convex subset Ω of Euclidean n-space, we consider the problem of minimizing the perimeter of subsets of Ω subject to a volume constraint. The problem is to determine whether in general a minimizer is also convex. Although this problem is unresolved, we show that if Ω satisfies a “great circle” condition, then any minimizer is convex. We say that Ω satisfies a great circle condition if the largest closed ball B contained in Ω has a great circle that is contained in the boundary of Ω. A great circle of B is defined as the intersection of the boundary of B with a hyperplane passing through the center of B.     

Manifolds whose curvature operator has constant eigenvalues at the basepoint

Osserman conjectured that if the curvature operatorR of a Riemannian manifoldM has constant eigenvalues, thenM is locally a rank-1 symmetric space or is flat. The pointwise question is considerably more complicated. We present examples of Riemannian manifolds so thatR has constant eigenvalues at the basepoint, butR is not the curvature operator of a rank-1 symmetric space. Research partially supported by the NSF and IHES.     

The structure of area-minimizing double bubbles

We show that the least area required to enclose two volumes in ℝⁿ orS ⁿ forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝⁿ have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝⁿ. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝⁿ, based on an idea due to Brian White. Supported in part by NSF DMS-9409166.     

Blow-up of optimal sets in the irrigation problem

We consider the minimization problem for an average distance functional in the plane, among all compact connected sets of prescribed length. For a minimizing set, the blow-up sequence in the neighborhood of any point is investigated. We show existence of the blow up limits and we characterize them, using the results to get some partial regularity statements.     

Global homeomorphisms and covering projections on metric spaces

For a large class of metric spaces with nice local structure, which includes Banach–Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path- liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of these sufficient conditions are also necessary. Finally, we give an application to the existence of global implicit functions. O. Gutú and J. A. Jaramillo were supported in part by D.G.E.S. (Spain) Grant BFM2003-06420.     

Regularity of area-minimizing surfaces in 3D polytopes and of invariant hypersurfaces inR n

In (the surface of) a convex polytope P ³ inR ⁴,an areaminimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an areaminimizing G-invariant oriented hypersurface is smooth (except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also treat other categories of surfaces such as rectifiable currents modulo v and soap films.     

A second order minimality condition for the Mumford-Shah functional

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on H ¹ ₀(Γ), Γ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of Γ. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.     

Gromov hyperbolicity through decomposition of metrics spaces II

In this article we study the hyperbolicity in the Gromov sense of metric spaces. We deduce the hyperbolicity of a space from the hyperbolicity of its “building block components,” which can be joined following an arbitrary scheme. These results are especially valuable since they simplify notably the topology and allow to obtain global results from local information. Some interesting theorems about the role of punctures and funnels on the hyperbolicity of Riemann surfaces can be deduced from the conclusions of this article.