Majorization by hemispheres and quadratic isoperimetric constants

Creutz, Paul

doi:10.1090/tran/7827

Majorization by hemispheres and quadratic isoperimetric constants

Author: Paul Creutz
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
Published electronically: November 12, 2019
MathSciNet review: 4068274
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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 https://doi.org/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

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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0377-0427%2893%29E0213-6

Dominic Descombes and Urs Lang, Convex geodesic bicombings and hyperbolicity, Geom. Dedicata 177 (2015), 367–384. MR 3370039, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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

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 https://doi.org/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 https://doi.org/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 https://doi.org/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

Cédric Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454