Université de Fribourg

Canonical parameterizations of metric disks

Lytchak, Alexander ; Wenger, Stefan

In: Duke Mathematical Journal, 2020, vol. 169, no. 4, p. 761–797

We use the recently established existence and regularity of area and energy minimizing disks in metric spaces to obtain canonical parameterizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parameterizations of linearly locally connected, Ahlfors 2-regular metric 2-spheres....

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

Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces

Bernig, Andreas ; Lytchak, Alexander

In: Journal für die reine und angewandte Mathematik, 2007, no. 608, p. 1-15

It is shown that the Gromov-Hausdorff limit of a subanalytic 1-parameter family of compact connected sets (endowed with the inner metric) exists. If the family is semialgebraic, then the limit space can be identified with a semialgebraic set over some real closed field. Different notions of tangent cones (pointed Gromov-Hausdorff limits, blow-ups and Alexandrov cones) for a closed connected...