The Douglas problem for parametric double integrals

 Let be a parametric variational double integral and γ ⊂ ℝ ⁿ be a system of several distinct Jordan curves. We prove the existence of multiply connected, conformally parametrized minimizers of spanned in γ by solving the Douglas problem for parametric functionals on multiply connected schlicht domains. As a by-product we obtain a simple isoperimetric inequality for multiply connected -minimizers, and we discuss regularity results up to the boundary which follow from corresponding results for the Plateau problem. Received: 19 April 2002 Mathematics Subject Classification (2000): 49J45, 49Q10, 53A07, 53A10     

Enclosure theorems for extremals of elliptic parametric functionals

In the following paper we study parametric functionals. First we introduce a generalized mean curvature (so called F-mean curvature). This enables us to describe extremals of parametric funcionals as surfaces of prescribed F-mean curvature. Furthermore we give a differential equation for arbitrary immersions generalizing and apply this equation to surfaces of vanishing and prescribed F-mean curvature. Especially we prove non-existence results for such surfaces generalizing Theorems by Hildebrandt and Dierkes [3], [6]. Received: 11 May 2001 / Accepted: 11 July 2001 / Published online: 12 October 2001     

On the isoperimetric problem in the Heisenberg group ℍ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

It has been recently conjectured that, in the context of the Heisenberg group ℍⁿ endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of the ℍ-perimeter among sets of constant Haar measure) could coincide with the solutions to a “restricted” isoperimetric problem within the class of sets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties of these restricted isoperimetric sets, which we call Heisenberg bubbles. In particular, we show that their boundary has constant mean ℍ-curvature and, quite surprisingly, that it is foliated by the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg bubbles are actually isoperimetric among the whole class of measurable subsets of ℍⁿ, we turn our attention to the relationship between volume, perimeter, and ε-enlargements. In particular, we prove a Brunn–Minkowski inequality with topological exponent as well as the fact that the ℍ-perimeter of a bounded, open set F⊂ℍⁿ of class C² can be computed via a generalized Minkowski content, defined by means of any bounded set whose horizontal projection is the 2n-dimensional unit disc. Some consequences of these properties are discussed. Mathematics Subject Classification (2000) 28A75, 22E25, 49Q20     

Finite order q-invariants of immersions of surfaces into 3-space

Given a surface F, we are interested in valued invariants of immersions of F into , which are constant on each connected component of the complement of the quadruple point discriminant in . Such invariants will be called “q-invariants.” Given a regular homotopy class , we denote by the space of all q-invariants on A of order . We show that ifF is orientable, then for each regular homotopy class A and each n, $\dim (V_n (A) / V_{n-1}(A) ) \leq 1$. Received June 15, 1999; in final form September 22, 1999 / Published online October 30, 2000     

Anchored expansion and random walk

This paper studies anchored expansion, a non-uniform version of the strong isoperimetric inequality. We show that every graph with i-anchored expansion contains a subgraph with isoperimetric (Cheeger) constant at least i. We prove a conjecture by Benjamini, Lyons and Schramm (1999) that in such graphs the random walk escapes with a positive lim inf speed. We also show that anchored expansion implies a heat-kernel decay bound of order exp(—cn ^1/3). Submitted: September 1999, Revision: January 2000.     

On the cases of equality in Bobkov's inequality and Gaussian rearrangement

We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the Faber-Krahn inequality. Received June 21, 1999 / Accepted June 12, 2000 / Published online November 9, 2000     

Quadratic isoperimetric inequality for mapping tori of polynomially growing automorphisms of free groups

We prove that the mapping torus F_n \rtimes_f \Bbb Z F_n \rtimes_\phi {\Bbb Z} of a polynomially growing automorphism f: F_n ? F_n \phi : F_n \to F_n of finitely generated free group F_n satisfies the quadratic isoperimetric inequality.     

Order one invariants of immersions of surfaces into 3-space

We classify all order one invariants of immersions of a closed orientable surface F into ³, with values in an arbitrary Abelian group . We show that for any F and and any regular homotopy class of immersions of F into ³, the group of all order one invariants on is isomorphic to is the group of all functions from a set of cardinality . Our work includes foundations for the study of finite order invariants of immersions of a closed orientable surface into ³, analogous to chord diagrams and the 1-term and 4-term relations of knot theory.Partially supported by the Minerva FoundationMathamatics Subject Classification (2000):57M, 57R42     

A characterization of diameter-2-critical graphs with no antihole of length four

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n ²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.     

Isoperimetric inequality from the poisson equation via curvature

In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q‐regular measure, where Q > 1, that supports a local L²‐Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L^∞‐norm of the gradient Du can be bounded by the Lorentz norm L^Q,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on $\||Du|\|_{L^\infty_{\rm loc}}$ . © 2011 Wiley Periodicals, Inc.     

Isoperimetric inequalities and conjugate points on Lorentzian surfaces

Recently, an isoperimetric inequality for a sector on the Minkowski 2-spacetime has been derived by the method of parallels and the relativistic Gauss-Bonnet formula. In the present paper, we derive an isoperimetric inequality for a sector on a Lorentzian surface with curvatureK ≤ C. As a sector can be modeled by a geodesic variation of a timelike geodesic, our isoperimetric inequality gives an upper bound for the spacelike boundary of a sector. As an application of our results, we give an elementary proof of the existence of conjugate points on a Lorentzian surface with curvatureK ≤ C < 0 and we obtain an upper bound for the (timelike) diameter of a globally hyperbolic Lorentzian surface withK ≤ C < 0 by comparison of sectors.     

A sharp isoperimetric inequality for rank one symmetric spaces

 We prove an isoperimetric inequality for compact, regular domains in rank one symmetric spaces, which is sharp for geodesic balls. Besides volume and area of a given domain, some weak information about the second fundamental form of its boundary is involved. Received: 2 September 2002 / Revised version: 10 December 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 53C35, 52A40, 51M25     

Some Isoperimetric Inequalities for Kernels of Free Extensions

If G is a hyperbolic group (resp. synchronously or asynchronously automatic group) which can be expressed as an extension of a finitely presented group H by a finitely generated free group, then the normal subgroup H satisfies a polynomial isoperimetric inequality (resp. exponential isoperimetric inequality).     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L _p -Petty projection inequality and an affine isoperimetric inequality of Γ_− p K.     

Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities

Let VIP(F,C) denote the variational inequality problem associated with the mapping F and the closed convex set C. In this paper we introduce weak conditions on the mapping F that allow the development of a convergent cutting-plane framework for solving VIP(F,C). In the process we introduce, in a natural way, new and useful notions of generalized monotonicity for which first order characterizations are presented. Received: September 25, 1997 / Accepted: March 2, 1999?Published online July 20, 2000     

Integral curvature estimates for F-stable hypersurfaces

We consider immersed hypersurfaces in euclidean which are stable with respect to an elliptic parametric functional with integrand F = F(N) depending on normal directions only. We prove an integral curvature estimate provided that F is sufficiently close to the area integrand, extending the classical estimate of Schoen, Simon and Yau [19] for stable minimal hypersurfaces in , as well as the pointwise estimate of Simon [22] for F-minimizing hypersurfaces. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with F. As an application, we obtain a new Bernstein result for complete F-stable hypersurfaces of dimension .Received: 14 July 2003, Accepted: 13 September 2004, Published online: 10 December 2004Mathematics Subject Classification: 53C42, 49Q10, 35J60     

Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks

We prove an isoperimetric inequality for wreath products of Markov chains with variable fibers. We use isoperimetric inequalities for wreath products to estimate the return probability of random walks on infinite groups and graphs, drift of random loops, the expected value E(exp(−tR _n )), where R _n is the number of distinct sites, visited up to the moment n, and, more generally, (where L _z,n is the number of visits of z up to the moment n and F(x, y) is some non-negative function).     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

An elementary proof of the Loomis–Whitney theorem

Loomis and Whitney proved an inequality between volume and areas of projections of an open set in n-dimensional space related to the isoperimetric inequality. They reduced the problem to a combinatorial theorem proved by a repeated use of Hölder inequality. In this paper we prove a general inequality between real numbers which easily implies the combinatorial theorem of Loomis and Whitney.

     

Uniqueness of conformal metrics with prescribed total curvature in R^2

In this paper, we investigate the solution structure of solutions of where K(x) is a H?lder function in . For a given positive total curvature, we consider the problem of the uniqueness of solutions with this prescribed total curvature. We apply various methods such as the method of moving spheres and the isoperimetric inequality to show the uniqueness for several classes of K. Received December 15, 1998 / Accepted April 23, 1999