A Sobolev Poincar\'e type inequality for integral varifolds

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.Comment: v1: 27 pages, no figures; v2: replaced citations of the author's dissertation by proofs, material of sections 1 and 3 reorganised, slightly more general results in section 2, some remarks, some discussion and some references added, 40 pages, no figure

On the isoperimetric problem in the Heisenberg group \u210dn

It has been recently conjectured that, in the context of the Heisenberg groupHn endowed with its Carnot\u2013Carath\ue9odory metric and Haar measure, the isoperimetricsets (i.e., minimizers of the H-perimeter among sets of constant Haar measure) couldcoincide with the solutions to a \u201crestricted\u201d isoperimetric problem within the class ofsets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper,we derive new properties of these restricted isoperimetric sets, which we call Heisenbergbubbles. In particular, we show that their boundary has constant mean H-curvature and, quitesurprisingly, that it is foliated by the family of minimal geodesics connecting two specialpoints. In view of a possible strategy for proving that Heisenberg bubbles are actuallyisoperimetric among the whole class of measurable subsets of Hn, we turn our attentionto the relationship between volume, perimeter, and -enlargements. In particular, we provea Brunn\u2013Minkowski inequality with topological exponent as well as the fact that the Hperimeterof a bounded, open set F 82 Hn of class C2 can be computed via a generalizedMinkowski content, defined by means of any bounded set whose horizontal projection is the2n-dimensional unit disc. Some consequences of these properties are discussed

On a linear programming approach to the discrete Willmore boundary value problem and generalizations

We consider the problem of finding (possibly non connected) discrete surfaces spanning a finite set of discrete boundary curves in the three-dimensional space and minimizing (globally) a discrete energy involving mean curvature. Although we consider a fairly general class of energies, our main focus is on the Willmore energy, i.e. the total squared mean curvature Our purpose is to address the delicate task of approximating global minimizers of the energy under boundary constraints. The main contribution of this work is to translate the nonlinear boundary value problem into an integer linear program, using a natural formulation involving pairs of elementary triangles chosen in a pre-specified dictionary and allowing self-intersection. Our work focuses essentially on the connection between the integer linear program and its relaxation. We prove that: - One cannot guarantee the total unimodularity of the constraint matrix, which is a sufficient condition for the global solution of the relaxed linear program to be always integral, and therefore to be a solution of the integer program as well; - Furthermore, there are actually experimental evidences that, in some cases, solving the relaxed problem yields a fractional solution. Due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design ad-hoc integer solvers that yield high-definition approximations to solutions of several boundary value problems involving mean curvature, in particular the Willmore boundary value problem

Instabilities of cylindrical bubble clusters

Small bubbles in an experimental two-dimensional foam between glass plates regularly undergo a three-dimensional instability as the small bubbles shrink under diffusion or equivalently as the plate separation increases, and end up on one of the plates. The most recent experiments of Cox, Weaire, and Vaz are accompanied by Surface Evolver computer simulations and rough theoretical calculations. We show how a recent second variation formula may be used to perform exact theoretical calculations for infinitesimal perturbations for such a system, and verify results with Surface Evolver simulations