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....
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In: Mathematische Annalen, 2019, vol. 373, no. 3, p. 1177–1210
We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are CAT(0) . This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov...
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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...
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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...
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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...
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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...
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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....
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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...
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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....
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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...
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