Isoperimetric estimates in products

In a product M ₁ × M ₂ of Riemannian manifolds, the least perimeter required to enclose given volume among general regions is at least 1/√ 2 times that among regions of product form, assuming that the isoperimetric profiles of M ₁ and M ₂ are concave. This result sharpens earlier work of Grigor'yan, generalizes results of Bollobás and Leader and of Barthe, and yields a lower bound on the Cheeger isoperimetric constant of a product.     

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.

     

On exactness and isoperimetric profiles of discrete groups

We introduce a quasi-isometry invariant related to Property A and explore its connections to various other invariants of finitely generated groups. This allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other. We apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory.     

On isoperimetric problems for domains with partly known boundaries

In the present paper, we discuss the isoperimetric problems for domains with partly known boundaries, i.e., the problem of determining a domain that minimizes the capacity functional in the class of plain double-connected domains having the same fixed area and outer boundary. The formulas for capacity variations obtained in this paper allows us to formulate necessary conditions.It is proved that the convexity of the fixed outer boundary implies the convexity of the inner boundary corresponding to an optimal domain. Then, we discuss the case where the fixed part of the boundary is a square.Further, we consider similar problems with more complicated functionals. We introduce the concept of a minimal function in the class of equimeasurable functions. This concept allows us to unify the approach to all of these problems. At the end, we produce a hypothesis that, if proved, would enable us to characterize the shape of the optimal domains in the isoperimetric problems mentioned above.The author wishes to express his appreciation to Dr. K. A. Lurie for his help and unceasing attention.     

Infinite Dimensional Isoperimetric Inequalities in Product Spaces with the Supremum Distance

We study the isoperimetric problem for product probability measures with respect to the uniform enlargement. We construct several examples of measures for which the isoperimetric function of coincides with the one of the infinite product . This completes earlier works by Bobkov and Houdré.     

A New Notion of Conjugacy for Isoperimetric Problems

For problems in the calculus of variations with isoperimetric side constraints, we provide in this paper a set of points whose emptiness, independently of nonsingularity assumptions, is equivalent to the nonnegativity of the second variation along admissible variations. The main objective of introducing a characterization of this condition should be, of course, to obtain a simpler way of verifying it. There are two other sets of points available in the literature, introduced by Loewen and Zheng (1994) and Zeidan (1996), for which this necessary condition implies their emptiness. However, we show that verifying membership of these sets may be more difficult than checking directly if that condition holds. Contrary to this behavior, we prove that the desired objective of characterizing that condition is achieved by means of the set introduced in this paper.     

Isoperimetric inequalities in crystallography

Given a cubic space group (viewed as a finite group of isometries of the torus ), we obtain sharp isoperimetric inequalities for -invariant regions. These inequalities depend on the minimum number of points in an orbit of and on the largest Euler characteristic among nonspherical -symmetric surfaces minimizing the area under volume constraint (we also give explicit estimates of this second invariant for the various crystallographic cubic groups ). As an example, we prove that any surface dividing into two equal volumes with the same (orientation-preserving) symmetries as the A. Schoen minimal Gyroid has area at least (the conjectured minimizing surface in this case is the Gyroid itself whose area is ).

     

Minimal webs in Riemannian manifolds

For a given combinatorial graph G a geometrization (G, g) of the graph is obtained by considering each edge of the graph as a 1-dimensional manifold with an associated metric g. In this paper we are concerned with minimal isometric immersions of geometrized graphs (G, g) into Riemannian manifolds (N ⁿ , h). Such immersions we call minimal webs. They admit a natural ‘geometric’ extension of the intrinsic combinatorial discrete Laplacian. The geometric Laplacian on minimal webs enjoys standard properties such as the maximum principle and the divergence theorems, which are of instrumental importance for the applications. We apply these properties to show that minimal webs in ambient Riemannian spaces share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in such spaces. In particular we use appropriate versions of the divergence theorems together with the comparison techniques for distance functions in Riemannian geometry and obtain bounds for the first Dirichlet eigenvalues, the exit times and the capacities as well as isoperimetric type inequalities for so-called extrinsic R-webs of minimal webs in ambient Riemannian manifolds with bounded curvature.      

Isoperimetry of group actions

The isoperimetric profile of a discrete group was introduced by Vershik, however it is well defined only for a restrictive class amenable groups. We generalize the notion of isoperimetric profile beyond the world of amenable groups by defining isoperimetric profiles of amenable actions of finitely generated groups on compact topological spaces. This allows to extend the definition of the isoperimetric profile to all groups which are exact in such a way that for amenable groups it is equal to Vershik's isoperimetric profile. The main feature of our construction is that it preserves many of the properties known from the classical case. We use these results to compute exact asymptotics of the isoperimetric profiles for several classes of non-amenable groups.