Morrey’s 𝜖-conformality lemma in metric spaces

Fitzi, Martin; Wenger, Stefan

doi:10.1090/proc/15065

Morrey’s $\varepsilon$-conformality lemma in metric spaces
HTML articles powered by AMS MathViewer

by Martin Fitzi and Stefan Wenger

Proc. Amer. Math. Soc. 148 (2020), 4285-4298

DOI: https://doi.org/10.1090/proc/15065

Published electronically: June 4, 2020

PDF | Request permission

Abstract:

We provide a simpler proof and slight strengthening of Morrey’s famous lemma on $\varepsilon$-conformal mappings. Our result more generally applies to Sobolev maps with values in a complete metric space, and we obtain applications to the existence of area minimizing surfaces of higher genus in metric spaces. Unlike Morrey’s proof, which relies on the measurable Riemann mapping theorem, we only need the existence of smooth isothermal coordinates established by Korn and Lichtenstein.

References

Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141
Luigi Ambrosio, Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 3, 439–478. MR 1079985
Kari Astala, Tadeusz Iwaniec, and Gaven Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009. MR 2472875
R. Courant, The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math. 72 (1940), 51–98. MR 2478, DOI 10.1007/BF02546328
Jesse Douglas, Minimal surfaces of higher topological structure, Ann. of Math. (2) 40 (1939), no. 1, 205–298. MR 1503457, DOI 10.2307/1968552
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Martin Fitzi and Stefan Wenger, Area minimizing surfaces of bounded genus in metric spaces, J. Reine Angew. Math., to appear.
Piotr Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no. 4, 403–415. MR 1401074, DOI 10.1007/BF00275475
Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson, Sobolev spaces on metric measure spaces, New Mathematical Monographs, vol. 27, Cambridge University Press, Cambridge, 2015. An approach based on upper gradients. MR 3363168, DOI 10.1017/CBO9781316135914
S. V. Ivanov, Volumes and areas of Lipschitz metrics, Algebra i Analiz 20 (2008), no. 3, 74–111 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 3, 381–405. MR 2454453, DOI 10.1090/S1061-0022-09-01053-X
Jürgen Jost, Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds, J. Reine Angew. Math. 359 (1985), 37–54. MR 794798, DOI 10.1515/crll.1985.359.37
Jürgen Jost, Generalized Dirichlet forms and harmonic maps, Calc. Var. Partial Differential Equations 5 (1997), no. 1, 1–19. MR 1424346, DOI 10.1007/s005260050056
M. B. Karmanova, Area and co-area formulas for mappings of the Sobolev classes with values in a metric space, Sibirsk. Mat. Zh. 48 (2007), no. 4, 778–788 (Russian, with Russian summary); English transl., Siberian Math. J. 48 (2007), no. 4, 621–628. MR 2355373, DOI 10.1007/s11202-007-0064-7
Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113–123. MR 1189747, DOI 10.1090/S0002-9939-1994-1189747-7
Nicholas J. Korevaar and Richard M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659. MR 1266480, DOI 10.4310/CAG.1993.v1.n4.a4
Arthur Korn, Zwei Anwendungen der Methode der sukzessiven Annäherungen, Schwarz Festschrift, Berlin, 1919, pp. 215–229.
Leon Lichtenstein, Zur Theorie der konformen Abbildungen; Konforme Abbildungen nicht-analytischer singlaritätenfreier Flächenstücke auf ebene Gebiete, Bull. Acad. Sci. Cracovie, 1916, pp. 192–217.
Alexander Lytchak and Stefan Wenger, Area minimizing discs in metric spaces, Arch. Ration. Mech. Anal. 223 (2017), no. 3, 1123–1182. MR 3594354, DOI 10.1007/s00205-016-1054-3
Alexander Lytchak and Stefan Wenger, Energy and area minimizers in metric spaces, Adv. Calc. Var. 10 (2017), no. 4, 407–421. MR 3707085, DOI 10.1515/acv-2015-0027
Charles B. Morrey Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. MR 1501936, DOI 10.1090/S0002-9947-1938-1501936-8
Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807–851. MR 27137, DOI 10.2307/1969401
Yu. G. Reshetnyak, Sobolev classes of functions with values in a metric space, Sibirsk. Mat. Zh. 38 (1997), no. 3, 657–675, iii–iv (Russian, with Russian summary); English transl., Siberian Math. J. 38 (1997), no. 3, 567–583. MR 1457485, DOI 10.1007/BF02683844
Yu. G. Reshetnyak, On the theory of Sobolev classes of functions with values in a metric space, Sibirsk. Mat. Zh. 47 (2006), no. 1, 146–168 (Russian, with Russian summary); English transl., Siberian Math. J. 47 (2006), no. 1, 117–134. MR 2215302, DOI 10.1007/s11202-006-0013-x
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
Max Shiffman, The Plateau problem for minimal surfaces of arbitrary topological structure, Amer. J. Math. 61 (1939), 853–882. MR 468, DOI 10.2307/2371631
Friedrich Tomi and Anthony J. Tromba, Existence theorems for minimal surfaces of nonzero genus spanning a contour, Mem. Amer. Math. Soc. 71 (1988), no. 382, iv+83. MR 920962, DOI 10.1090/memo/0382