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Trapping quasiminimizing submanifolds in spaces of negative curvature

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Commentarii Mathematici Helvetici

Abstract

LetM be a Hadamard manifold with all sectional curvatures bounded above by some negative constant. A well-known lemma essentially due to M. Morse states that every quasigeodesic segment inM lies within an a priori bounded distance from the geodesic arc connecting its endpoints. In this paper we establish an analogue of this fact for quasiminimizing surfaces in all dimensions and codimensions; the only additional requirement is that the sectional curvatures ofM be bounded from below as well. We apply this estimate to obtain new solutions to the asymptotic Plateau problem in various settings.

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Authors and Affiliations

  1. Mathematisches Institut, Hebelstr. 29, D-79104, Freiburg, Germany

    Victor Bangert

  2. Forschungsinstitut für Mathematik, ETH Zentrum, CH-8092, Zürich, Switzerland

    Urs Lang

Authors
  1. Victor Bangert
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  2. Urs Lang
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Additional information

The second author was supported by the Swiss National Science Foundation and enjoyed the hospitality of the University of Bonn. The collaboration between the authors was facilitated by the program GADGET II.

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Bangert, V., Lang, U. Trapping quasiminimizing submanifolds in spaces of negative curvature. Commentarii Mathematici Helvetici 71, 122–143 (1996). https://doi.org/10.1007/BF02566412

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  • DOI: https://doi.org/10.1007/BF02566412

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