arXiv subjects

Stefan Wenger

Publications and source records attributed to Stefan Wenger.

At least 37 records · Page 2Linked to original sources

arXiv2015

Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hoelder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.

math.AP↗

arXiv2015

Energy and area minimizers in metric spaces

We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hoelder exponents of some area-minimizing discs.

math.DG↗

arXiv2015

Area minimizing discs in metric spaces

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.

math.DG↗

arXiv2014

Wolfe's theorem for weakly differentiable cochains

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension $m$ in $\mathbb{R}^n$ with the space of flat $m$-cochains, that is, the dual space of flat chains of dimension $m$ in $\mathbb{R}^n$. The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in $\mathbb{R}^n$. A suitable theory of Sobolev cochains has recently been initiated by the second and third author. It is based on the concept of upper norm and upper gradient of a cochain, introduced in analogy with Heinonen-Koskela's concept of upper gradient of a function.

math.AP↗

arXiv2013

Lipschitz homotopy groups of the Heisenberg groups

Lipschitz and horizontal maps from an $n$-dimensional space into the $(2n+1)$-dimensional Heisenberg group $\H^n$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from $S^k$ to $\H^n$ which factor through $n$-spheres and showed that these maps have no smooth horizontal fillings. In this paper, however, we build on an example of Kaufman to show that these maps sometimes have Lipschitz fillings. This shows that the Lipschitz and the smooth horizontal homotopy groups of a space may differ. Conversely, we show that any Lipschitz map $S^k\to \H^1$ factors through a tree and is thus Lipschitz null-homotopic if $k\ge 2$.

math.GT↗

arXiv2012

An upper gradient approach to weakly differentiable cochains

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchheim's theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen-Koskela's concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with $p$-integrable upper gradient in $n$-dimensional Lie groups endowed with a left-invariant Finsler metric. Our result generalizes the well-known Morrey-Sobolev inequality for Sobolev functions. Finally, we prove several results relating capacity and modulus to Hausdorff dimension.

math.AP↗

arXiv2012

Plateau's problem for integral currents in locally non-compact metric spaces

The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak$^*$-compactness theorem for integral currents in dual spaces of separable Banach spaces. Our theorems generalize results of Ambrosio-Kirchheim, Lang, the author, and recent results of Ambrosio-Schmidt.

math.DG↗

arXiv2010

Nilpotent groups without exactly polynomial Dehn function

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than $n^2\log n$. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.

math.GR↗

arXiv2010

The pointed flat compactness theorem for locally integral currents

Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio--Kirchheim theory of currents in metric spaces, including currents with finite mass in bounded sets.

math.DG↗

arXiv2009

Lipschitz extensions into Jet space Carnot groups

The aim of this article is to prove a Lipschitz extension theorem for partially defined Lipschitz maps to jet spaces endowed with a left-invariant sub-Riemannian Carnot-Carathéodory distance. The jet spaces give a model for a certain class of Carnot groups, including in particular all Heisenberg groups. The proofs or our theorems are rather elementary, they are inspired by ideas in papers of Gromov, Young, and Lang-Schlichenmaier.

math.MG↗

arXiv2009

Lipschitz non-extension theorems into jet space Carnot groups

We prove non-extendability results for Lipschitz maps with target space being jet spaces equipped with a left-invariant Riemannian distance, as well as jet spaces equipped with a left-invariant sub-Riemannian Carnot-Caratheodory distance. The jet spaces give a model for a certain class of Carnot groups, including in particular all Heisenberg groups.

math.MG↗

arXiv2009

Rectifiability of flat chains in Banach spaces with coefficients in $Z_p$

Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and with respect to weaker mass bounds. A crucial tool in many proofs is the isoperimetric inequality proved by Ambrosio-Katz with universal constants.

math.CA↗

arXiv2009

Weak convergence and cancellation

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in terms of topology of the supports which ensure that no cancellation occurs and that the support of the weak limit agrees with the Hausdorff limit of the supports. We use our results to prove countable $\hm^m$-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds $M_n$ all of which are $λ$-linearly locally contractible up to some scale $r_0$. In an appendix, written by Raanan Schul and the second author, it is shown that the Gromov-Hausdorff limit need not be countably $\hm^m$-rectifiable if the $M_n$ have a common local geometric contractibility function which is only concave (and not linear). We also relate our results to work of Cheeger-Colding on the limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature.

math.DG↗

arXiv2008

Compactness for manifolds and integral currents with bounded diameter and volume

By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented $k$-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio-Kirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which we replace uniform compactness of the sequence with uniform bounds on volume and diameter.

math.DG↗

arXiv2008

The asymptotic rank of metric spaces

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a proper, cocompact, simply-connected geodesic metric space of non-curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.

math.DG↗

arXiv2007

Characterizations of metric trees and Gromov hyperbolic spaces

In a recent paper Chatterji and Niblo proved that a geodesic metric space is Gromov hyperbolic if and only if the intersection of any two closed balls has uniformly bounded eccentricity. In their paper, the authors raise the question whether a geodesic metric space with the property that the intersection of any two closed balls has eccentricity 0, is necessarily a real tree. The purpose of this note is to answer this question affirmatively. We also partially improve the main result of Chatterji and Niblo by showing that already sublinear eccentricity implies Gromov hyperbolicity.

math.MG↗

arXiv2007

A short proof of Gromov's filling inequality

We give a very short and rather elementary proof of Gromov's filling volume inequality for n-dimensional Lipschitz cycles (with integer and Z_2-coefficients) in $L^\infty$-spaces. This inequality is used in the proof of Gromov's systolic inequality for closed aspherical Riemannian manifolds and is often regarded as the difficult step therein.

math.DG↗

arXiv2007

Gromov hyperbolic spaces and the sharp isoperimetric constant

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our results strengthen and generalize theorems of Gromov, Papasoglu and others.

math.DG↗