Journal article

Spaces with almost Euclidean Dehn function

  • Wenger, Stefan Department of Mathematics, University of Fribourg, Switzerland
Published in:
  • Mathematische Annalen. - 2019, vol. 373, no. 3, p. 1177–1210
English 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 hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper CAT(0) in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being CAT(0) up to that scale.
Faculté des sciences et de médecine
Département de Mathématiques
  • English
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