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Cheeger constants of surfaces and isoperimetric inequalities


Author: Panos Papasoglu
Journal: Trans. Amer. Math. Soc. 361 (2009), 5139-5162
MSC (2000): Primary 53C20, 53C23, 20F65
DOI: https://doi.org/10.1090/S0002-9947-09-04815-6
Published electronically: May 19, 2009
MathSciNet review: 2515806
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that if the isoperimetric profile of a bounded genus non-compact surface grows faster than $ \sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces.

We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 it is sub-Euclidean and there is a $ g$ such that minimizers in dimension 3 have genus at most $ g$, then the filling function in dimension 3 is `almost' linear.


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  • 1. F. J. Almgren, Jr., An isoperimetric inequality, Proc. Amer. Math. Soc. 15 (1964), 284-285. MR 0159925 (28:3141)
  • 2. A. S. Besicovitch, On two problems of Loewner, J. London Math. Soc. 27 (1952), 141-144. MR 0047126 (13:831d)
  • 3. M. Bonk, A. Eremenko, Uniformly hyperbolic surfaces, Indiana Univ. Math. J. 49 (2000), no. 1, 61-80. MR 1777037 (2001g:53120)
  • 4. N. Brady, M. R. Bridson, There is only one gap in the isoperimetric spectrum, Geom. Funct. Anal. 10 (2000), no. 5, 1053-1070. MR 1800063 (2001j:20046)
  • 5. N. Brady, M. R. Bridson, M. Forester, K. Shankar, Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra, preprint.
  • 6. M. R. Bridson, Fractional isoperimetric inequalities and subgroup distortion, J. Amer. Math. Soc. 12 (1999), no. 4, 1103-1118. MR 1678924 (2001a:20062)
  • 7. I. Benjamini, J. Cao, A new isoperimetric theorem for surfaces of variable curvature, Duke Math. J. 85 (1996), 359-396. MR 1417620 (97m:58046)
  • 8. B. H. Bowditch, A short proof that a subquadratic isoperimetric inequality implies a linear one, Michigan Math. J. 42 (1995), no. 1, 103-107. MR 1322192 (96b:20046)
  • 9. J. Burillo, J. Taback, Equivalence of geometric and combinatorial Dehn functions, New York J. Math. 8 (2002), 169-179. MR 1934388 (2004c:20070)
  • 10. M. Coornaert, T. Delzant, A. Papadopoulos, Geometrie et theorie des groupes, Lecture Notes in Mathematics, 1441, Springer-Verlag, Berlin, 1990. x+165 pp. MR 1075994 (92f:57003)
  • 11. C. B. Croke, A sharp four-dimensional isoperimetric inequality., Comment. Math. Helv. 59 (1984), no. 2, 187-192. MR 749103 (85f:53060)
  • 12. C. Drutu, Cones asymptotiques et invariants de quasi-isometrie pour des espaces metriques hyperboliques, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 1, 81-97. MR 1821069 (2002h:53069)
  • 13. D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, W. P. Thurston,Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp. MR 1161694 (93i:20036)
  • 14. S. M. Gersten, Subgroups of word hyperbolic groups in dimension $ 2$, J. London Math. Soc. (2) 54 (1996), no. 2, 261-283. MR 1405055 (97e:20058)
  • 15. J. R. Gilbert, J. P. Hutchinson, R. E. Tarjan, A separator theorem for graphs of bounded genus J. Algorithms 5 (1984), 391-407. MR 756165 (86h:68145)
  • 16. R. Grimaldi, P. Pansu, Remplissage et surfaces de revolution, J. Math. Pures Appl. (9) 82 (2003), no. 8, 1005-1046. MR 2005203 (2004f:53032)
  • 17. M. Gromov, Filling Riemannian manifolds J. Differential Geom. 18 (1983), no. 1, 1-147. MR 697984 (85h:53029)
  • 18. M. Gromov, Hyperbolic groups, Essays in group theory (S. M. Gersten, ed.), MSRI Publ. 8, Springer-Verlag, 1987, 75-263. MR 919829 (89e:20070)
  • 19. M. Gromov, Asymptotic invariants of infinite groups in `Geometric group theory' (G. Niblo, M. Roller, Eds.), LMS Lecture Notes, vol. 182, Cambridge Univ. Press, 1993. MR 1253544 (95m:20041)
  • 20. M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, with appendices by M. Katz, P. Pansu and S. Semmes, Progress in Mathematics, 152, Birkhauser Boston, Inc., Boston, MA, 1999. xx+585 pp. MR 1699320 (2000d:53065)
  • 21. A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. xii+544 pp. MR 1867354 (2002k:55001)
  • 22. J. Hersch, Quatre proprietes isoperimetriques de membranes spheriques homogenes, C.R. Acad. Sci. Paris 270 (1970), 1645-1648. MR 0292357 (45:1444)
  • 23. B. Kleiner, An isoperimetric comparison theorem, Invent. Math. 108 (1992), no. 1, 37-47. MR 1156385 (92m:53056)
  • 24. N. Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Diff. Geom. 37 (1993), 73-93. MR 1198600 (94d:58153)
  • 25. R. J. Lipton, R. E. Tarjan, Applications of a planar separator theorem, SIAM J. Comput. 9 (1980), 615-627. MR 584516 (82e:68067)
  • 26. F. Morgan, M. Hutchings, H. Howards, The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature, Trans. Amer. Math. Soc. 352 (2000), no. 11, 4889-4909. MR 1661278 (2001b:58024)
  • 27. A. Yu. Olshanski, Hyperbolicity of groups with subquadratic isoperimetric inequality, Internat. J. Algebra Comput. 1 (1991), no. 3, 281-289. MR 1148230 (93d:20067)
  • 28. P. Pansu, Une inégalité isopérimétrique pour le groupe d'Heisenberg, C.R. Acad. Sci. Paris 295 (1982), 127-131. MR 676380 (85b:53044)
  • 29. P. Papasoglu, An algorithm detecting hyperbolicity, Geometric and computational perspectives on infinite groups, 193-200, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 25, AMS, Providence, RI, 1996. MR 1364185 (96k:20075)
  • 30. P. Papasoglu, Isodiametric and isoperimetric inequalities for complexes and groups, J. London Math. Soc. (2) 62 (2000), no. 1, 97-106. MR 1771853 (2001h:20052)
  • 31. L. Polterovich, J-C. Sikorav, A linear isoperimetric inequality for the punctured Euclidean plane, preprint, arXive math.GR/0106216.
  • 32. M. Ritoré, The isoperimetric problem in complete surfaces of nonnegative curvature, J. Geom. Anal. 11 (2001), no. 3, 509-517. MR 1857855 (2002f:53109)
  • 33. M. Ritoré, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, Comm. Anal. Geom. 9 (2001), no. 5, 1093-1138. MR 1883725 (2003a:53018)
  • 34. M. V. Sapir, J-C. Birget, E. Rips, Isoperimetric and isodiametric functions of groups, Ann. of Math. (2) 156 (2002), no. 2, 345-466. MR 1933723 (2005b:20077a)
  • 35. P. Topping, Mean curvature flow and geometric inequalities, J. Reine Angew. Math. 503 (1998), 47-61. MR 1650335 (99m:53080)
  • 36. X. Wang, S. J. Pride, Second order Dehn functions and HNN-extensions, J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 272-288. MR 1717418 (2002k:20076)
  • 37. S. Wenger, Gromov hyperbolic spaces and the sharp isoperimetric constant, Invent. Math. 171 (2008), no. 1, 227-255. MR 2358060
  • 38. S. Wenger, Isoperimetric inequalities and the asymptotic rank of metric spaces, preprint.
  • 39. P. Yang, S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 55-63. MR 577325 (81m:58084)

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Additional Information

Panos Papasoglu
Affiliation: Department of Mathematics, University of Athens, Athens 157 84, Greece
Email: panos@math.uoa.gr

DOI: https://doi.org/10.1090/S0002-9947-09-04815-6
Received by editor(s): August 3, 2007
Published electronically: May 19, 2009
Additional Notes: This work was co-funded by the European Social Fund (75%) and the National Resources (25%) (Epeaek II) Pythagoras
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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