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Transactions of the American Mathematical Society

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



Area minimizing surfaces in homotopy classes in metric spaces

Authors: Elefterios Soultanis and Stefan Wenger
Journal: Trans. Amer. Math. Soc. 375 (2022), 4711-4739
MSC (2020): Primary 49Q05; Secondary 53A10, 53C23
Published electronically: April 21, 2022
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Abstract: 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

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

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
Article copyright: © Copyright 2022 American Mathematical Society