Dehn functions and Hölder extensions in asymptotic cones

Lytchak, Alexander; Wenger, Stefan; Young, Robert

doi:10.1515/crelle-2018-0041

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Journal article

Dehn functions and Hölder extensions in asymptotic cones

Lytchak, Alexander Mathematisches Institut, Universität Köln, Weyertal 86–90, 50931 Köln, Germany
Wenger, Stefan Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700 Fribourg, Switzerland
Young, Robert Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA

01.06.2020

Published in:

Journal für die reine und angewandte Mathematik. - 2020, vol. 2020, no. 763, p. 79–109

English 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 generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

Faculty

Faculté des sciences et de médecine

Department

Département de Mathématiques

Language