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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Area minimizing surfaces in homotopy classes in metric spaces
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by Elefterios Soultanis and Stefan Wenger PDF
Trans. Amer. Math. Soc. 375 (2022), 4711-4739 Request permission


We introduce and study a notion of relative $1$-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative $1$-homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively $1$-homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given $1$-homotopy class. Our theorems generalize and strengthen results of Lemaire [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), pp. 91–103], Jost [J. Reine Angew. Math. 359 (1985), pp. 37-54], Schoen–Yau [Ann. of Math. (2) 110 (1979), pp. 127–142], and Sacks–Uhlenbeck [Trans. Amer. Math. Soc. 271 (1982), pp. 639–652].
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Additional Information
  • Elefterios Soultanis
  • Affiliation: IMAPP, Radboud University, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands
  • MR Author ID: 989349
  • ORCID: 0000-0001-9514-3941
  • Email:
  • Stefan Wenger
  • Affiliation: Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700 Fribourg, Switzerland
  • MR Author ID: 764752
  • ORCID: 0000-0003-3645-105X
  • Email:
  • Received by editor(s): January 12, 2021
  • Received by editor(s) in revised form: June 15, 2021, and October 14, 2021
  • Published electronically: April 21, 2022
  • Additional Notes: The research was supported by Swiss National Science Foundation Grants 165848 and 182423
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4711-4739
  • MSC (2020): Primary 49Q05; Secondary 53A10, 53C23
  • DOI:
  • MathSciNet review: 4439489