Majorization by hemispheres and quadratic isoperimetric constants
HTML articles powered by AMS MathViewer
- by Paul Creutz PDF
- Trans. Amer. Math. Soc. 373 (2020), 1577-1596 Request permission
Abstract:
Let $X$ be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed $L$-Lipschitz curve $\gamma :S^1\rightarrow X$ may be extended to an $L$-Lipschitz map defined on the hemisphere $f:H^2\rightarrow X$. This implies that $X$ satisfies a quadratic isoperimetric inequality (for curves) with constant $\frac {1}{2\pi }$. We discuss how this fact controls the regularity of minimal discs in Finsler manifolds when applied to the work of Alexander Lytchak and Stefan Wenger.References
- Luigi Ambrosio and Daniele Puglisi, Linear extension operators between spaces of Lipschitz maps and optimal transport, J. Reine Angew. Math., 2019.
- Giuliano Basso, Fixed point theorems for metric spaces with a conical geodesic bicombing, Ergodic Theory Dynam. Systems 38 (2018), no. 5, 1642–1657. MR 3819996, DOI 10.1017/etds.2016.106
- Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
- D. Burago and S. Ivanov, On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume, Ann. of Math. (2) 156 (2002), no. 3, 891–914. MR 1954238, DOI 10.2307/3597285
- Dmitri Burago and Sergei Ivanov, Minimality of planes in normed spaces, Geom. Funct. Anal. 22 (2012), no. 3, 627–638. MR 2972604, DOI 10.1007/s00039-012-0170-y
- Giuliano Basso and Benjamin Miesch, Conical geodesic bicombings on subsets of normed vector spaces, Adv. Geom. (to appear).
- Sergei Buyalo and Viktor Schroeder, Extension of Lipschitz maps into 3-manifolds, Asian J. Math. 5 (2001), no. 4, 685–704. MR 1913816, DOI 10.4310/AJM.2001.v5.n4.a5
- Carlos A. Cabrelli and Ursula M. Molter, The Kantorovich metric for probability measures on the circle, J. Comput. Appl. Math. 57 (1995), no. 3, 345–361. MR 1335789, DOI 10.1016/0377-0427(93)E0213-6
- Dominic Descombes and Urs Lang, Convex geodesic bicombings and hyperbolicity, Geom. Dedicata 177 (2015), 367–384. MR 3370039, DOI 10.1007/s10711-014-9994-y
- Richard M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. MR 982264
- Siegfried Gähler and Grattan Murphy, A metric characterization of normed linear spaces, Math. Nachr. 102 (1981), 297–309. MR 642160, DOI 10.1002/mana.19811020125
- Chang-Yu Guo and Stefan Wenger, Area minimizing discs in locally non-compact metric spaces, Comm. Anal. Geom. (to appear), preprint, arXiv:1701.06736, 2017.
- J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. MR 182949, DOI 10.1007/BF02566944
- 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
- S. V. Ivanov, Filling minimality of Finslerian 2-discs, Tr. Mat. Inst. Steklova 273 (2011), no. Sovremennye Problemy Matematiki, 192–206; English transl., Proc. Steklov Inst. Math. 273 (2011), no. 1, 176–190. MR 2893545, DOI 10.1134/S0081543811040079
- 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
- Hans G. Kellerer, Duality theorems and probability metrics, Proceedings of the seventh conference on probability theory (Braşov, 1982) VNU Sci. Press, Utrecht, 1985, pp. 211–220. MR 867434
- 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
- Urs Lang, Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5 (2013), no. 3, 297–331. MR 3096307, DOI 10.1142/S1793525313500118
- James R. Lee and Assaf Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95. MR 2129708, DOI 10.1007/s00222-004-0400-5
- U. Lang, B. Pavlović, and V. Schroeder, Extensions of Lipschitz maps into Hadamard spaces, Geom. Funct. Anal. 10 (2000), no. 6, 1527–1553. MR 1810752, DOI 10.1007/PL00001660
- U. Lang and V. Schroeder, Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal. 7 (1997), no. 3, 535–560. MR 1466337, DOI 10.1007/s000390050018
- 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
- Alexander Lytchak and Stefan Wenger, Isoperimetric characterization of upper curvature bounds, Acta Math. 221 (2018), no. 1, 159–202. MR 3877021, DOI 10.4310/ACTA.2018.v221.n1.a5
- Alexander Lytchak, Stefan Wenger, and Robert Young, Dehn functions and Hölder extensions in asymptotic cones, J. Reine Angew. Math (to appear), preprint, arXiv:1608.00082, 2016.
- Alexander Lytchak and Asli Yaman, On Hölder continuous Riemannian and Finsler metrics, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2917–2926. MR 2216252, DOI 10.1090/S0002-9947-06-04195-X
- Kurt Mahler, Ein Minimalproblem für konvexe Polygone, Mathematica (Zutphen) B, 7 (1939), 118–127.
- Shin-ichi Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425. MR 2480975, DOI 10.1007/s11117-008-2202-2
- Patrick Overath and Heiko von der Mosel, Plateau’s problem in Finsler 3-space, Manuscripta Math. 143 (2014), no. 3-4, 273–316. MR 3167617, DOI 10.1007/s00229-013-0626-x
- Athanase Papadopoulos, Metric spaces, convexity and non-positive curvature, 2nd ed., IRMA Lectures in Mathematics and Theoretical Physics, vol. 6, European Mathematical Society (EMS), Zürich, 2014. MR 3156529, DOI 10.4171/132
- Sven Pistre and Heiko von der Mosel, The Plateau problem for the Busemann-Hausdorff area in arbitrary codimension, Eur. J. Math. 3 (2017), no. 4, 953–973. MR 3736793, DOI 10.1007/s40879-017-0163-3
- Ju. G. Rešetnjak, Non-expansive maps in a space of curvature no greater than $K$, Sibirsk. Mat. Ž. 9 (1968), 918–927 (Russian). MR 0244922
- 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
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
- Karl-Theodor Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390. MR 2039961, DOI 10.1090/conm/338/06080
- A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
- Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9
Additional Information
- Paul Creutz
- Affiliation: Mathematisches Institut der Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
- Email: pcreutz@math.uni-koeln.de
- Received by editor(s): October 30, 2018
- Received by editor(s) in revised form: February 6, 2019, and February 12, 2019
- Published electronically: November 12, 2019
- Additional Notes: The author was partially supported by the DFG grant SPP 2026.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 1577-1596
- MSC (2010): Primary 46B09, 53A10; Secondary 52A38, 53C60, 46B20
- DOI: https://doi.org/10.1090/tran/7827
- MathSciNet review: 4068274