Université de Fribourg

Intrinsic structure of minimal discs in metric spaces

Lytchak, Alexander ; Wenger, Stefan

In: Geometry & Topology, 2017, vol. 22, no. 1, p. 591–644

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology...

Université de Fribourg

Energy and area minimizers in metric spaces

Lytchak, Alexander ; Wenger, Stefan

In: Advances in Calculus of Variations, 2016, vol. 10, no. 4, p. 407–421

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...

Université de Fribourg

Area minimizing discs in metric spaces

Lytchak, Alexander ; Wenger, Stefan

In: Archive for Rational Mechanics and Analysis, 2017, vol. 223, no. 3, p. 1123–1182

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...

Université de Fribourg

Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

Lytchak, Alexander ; Wenger, Stefan

In: Calculus of Variations and Partial Differential Equations, 2016, vol. 55, no. 4, p. 98

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 Hölder continuity and continuity...

Université de Fribourg

Wolfe’s theorem for weakly differentiable cochains

Petit, Camille ; Rajala, Kai ; Wenger, Stefan

In: Journal of Functional Analysis, 2015, vol. 268, no. 8, p. 2261–2297

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in RⁿRn with the space of flat m-cochains, that is, the dual space of flat chains of dimension m in RⁿRn. The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in RⁿRn....

Université de Fribourg

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

Wenger, Stefan

In: Advances in Calculus of Variations, 2014, vol. 7, no. 2, p. 227–240

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...

Université de Fribourg

Lipschitz homotopy groups of the Heisenberg groups

Wenger, Stefan ; Young, Robert

In: Geometric and Functional Analysis, 2014, vol. 24, no. 1, p. 387–402

Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group Hn are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from S to Hn which factor through n-spheres and showed that these maps have no smooth horizontal fillings....

Université de Fribourg

An upper gradient approach to weakly differentiable cochains

Rajala, Kai ; Wenger, Stefan

In: Journal de Mathématiques Pures et Appliquées, 2013, vol. 100, no. 6, p. 868–906

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)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio–Kirchheimʼs theory of metric currents. The notion of weak differentiability we...