In: Calculus of Variations and Partial Differential Equations, 2015, vol. 52, no. 3-4, p. 469-488
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In: The Journal of Geometric Analysis, 2015, vol. 25, no. 4, p. 2590-2616
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In: Geometriae Dedicata, 2015, vol. 177, no. 1, p. 367-384
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In: Advances in Calculus of Variations, 2017, vol. 10, no. 4, p. 407-421
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In: Journal für die reine und angewandte Mathematik, 2020, vol. 2020, no. 763, p. 79–109
The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we...
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In: Calculus of Variations and Partial Differential Equations, 2020, vol. 59, no. 5, p. 177
We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by...
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In: Proceedings of the American Mathematical Society, 2020, vol. 148, no. 10, p. 4285–4298
We provide a simpler proof and slight strengthening of Morrey's famous lemma on $ \varepsilon $-conformal mappings. Our result more generally applies to Sobolev maps with values in a complete metric space, and we obtain applications to the existence of area minimizing surfaces of higher genus in metric spaces. Unlike Morrey's proof, which relies on the measurable Riemann mapping theorem, we...
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In: Bulletin of the London Mathematical Society, 2020, vol. 52, no. 3, p. 472–488
A theorem of Dorronsoro from the 1980s quantifies the fact that real‐valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last...
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In: European Journal of Combinatorics, 2020, vol. 87, p. 103098
We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik (2010).In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid.- We give explicit formulas for the face numbers of fundamental polytopes and ...
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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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