Stefan Wenger

Isoperimetric inequalities of Euclidean type in metric spaces

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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.

A short proof of Gromov’s filling inequality

We give a very short and rather elementary proof of Gromovs filling volume inequality for n-dimensional Lipschitz cycles (with integer and Z 2 -coefficients) in L∞-spaces. This inequality is used in the proof of Gromovs systolic inequality for closed aspherical Riemannian manifolds and is often regarded as the difficult step therein.

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 Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.

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 isoperimetric filling functions. For a proper, cocompact, simply connected geodesic metric space of non-positive curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.

Energy and area minimizers in metric spaces

Abstract We show that in the setting of proper metric spaces one obtains a solution of the classical 2-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area 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 Hölder exponents of some area-minimizing discs.

Filling invariants at infinity and the Euclidean rank of Hadamard spaces

In this paper we study a homological version of the asymptotic filling invariant divk defined by Brady and Farb in [BrFa] and show that it is a quasi-isometry invariant for all proper cocompact Hadamard spaces, i.e. proper cocompact CAT(0)-spaces, and that it can furthermore be used to detect the Euclidean rank of such spaces. We thereby extend results of [BrFa, Leu, Hin] from the setting of symmetric spaces of non-compact type to that of Hadamard spaces. Finally, we exhibit the optimal growth of the k-th homological divergence for symmetric spaces of non-compact type with Euclidean rank no larger than k and for CAT(κ)-spaces with κ < 0.

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

Abstract. 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.