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Quasi-isometries and isoperimetric inequalities in planar domains. (English) Zbl 1311.30016

Summary: This paper studies the stability of isoperimetric inequalities under quasi-isometries between non-exceptional Riemann surfaces endowed with their Poincaré metrics. This stability was proved by Kanai in the more general setting of Riemannian manifolds under the condition of positive injectivity radius. The present work proves the stability of the linear isoperimetric inequality for planar surfaces (genus zero surfaces) without any condition on their injectivity radii. It is also shown the stability of any non-linear isoperimetric inequality.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30F20 Classification theory of Riemann surfaces
53C20 Global Riemannian geometry, including pinching
31C12 Potential theory on Riemannian manifolds and other spaces
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[1] V. Alvarez, J. M. Rodríguez and D. V. Yakubovich, Estimates for nonlinear harmonic “measures” on trees, Michigan Math. J., 49 (2001), 47-64. · Zbl 1006.31006
[2] W. Ballman, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser, Boston, 1985.
[3] S. Bermudo, J. M. Rodríguez, J. M. Sigarreta and J.-M. Vilaire, Gromov hyperbolic graphs, Discrete Math., 313 (2013), 1575-1585. · Zbl 1279.05017
[4] L. Bers, An inequality for Riemann surfaces, In: Differential Geometry and Complex Analysis, A Volume Dedicated to the Memory of Harry Ernest Rauch, (eds. I. Chavel and H. M. Farkas), Springer-Verlag, 1985, pp.,87-93.
[5] C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math., 179 (1997), 1-39. · Zbl 0921.30032
[6] P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progr. Math., 106 , Birkhäuser, Boston, 1992. · Zbl 0770.53001
[7] A. Cantón, J. L. Fernández, D. Pestana and J. M. Rodríguez, On harmonic functions on trees, Potential Anal., 15 (2001), 199-244. · Zbl 1008.31006
[8] I. Chavel, Eigenvalues in Riemannian Geometry, Pure Appl. Math., 115 , Academic Press, New York, 1984.
[9] I. Chavel, Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives, Cambridge Tracts in Math., 145 , Cambridge University Press, Cambridge, 2001.
[10] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, In: Problems in Analysis, Princeton University, 1969, (ed. Robert C. Gunning), Princeton University Press, Princeton, 1970, pp.,195-199.
[11] J. L. Fernández and M. V. Melián, Bounded geodesics of Riemann surfaces and hyperbolic manifolds, Trans. Amer. Math. Soc., 347 (1995), 3533-3549. · Zbl 0845.30029
[12] J. L. Fernández and M. V. Melián, Escaping geodesics of Riemannian surfaces, Acta Math., 187 (2001), 213-236. · Zbl 1001.53025
[13] J. L. Fernández, M. V. Melián and D. Pestana, Quantitative mixing results and inner functions, Math. Ann., 337 (2007), 233-251. · Zbl 1125.30019
[14] J. L. Fernández, M. V. Melián and D. Pestana, Expanding maps, shrinking targets and hitting times, Nonlinearity, 25 (2012), 2443-2471. · Zbl 1256.37022
[15] J. L. Fernández and J. M. Rodríguez, The exponent of convergence of Riemann surfaces. Bass Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math., 15 (1990), 165-183. · Zbl 0702.30046
[16] J. L. Fernández and J. M. Rodríguez, Area growth and Green’s function of Riemann surfaces, Ark. Mat., 30 (1992), 83-92.
[17] É. Ghys and P. de la Harpe, Sur les Groupes Hyperboliques d’aprés Mikhael Gromov, Progr. Math., 83 , Birkhäuser, 1990.
[18] M. Gromov, Hyperbolic groups, In: Essays in Group Theory, (ed. S. M. Gersten), Math. Sci. Res. Inst. Publ., 8 , Springer-Verlag, 1987, pp.,75-263.
[19] I. Holopainen and P. M. Soardi, \(p\)-harmonic functions on graphs and manifolds, Manuscripta Math., 94 (1997), 95-110. · Zbl 0898.31007
[20] M. Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan, 37 (1985), 391-413. · Zbl 0554.53030
[21] M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan, 38 (1986), 227-238. · Zbl 0577.53031
[22] M. Kanai, Analytic inequalities, and rough isometries between non-compact Riemannian manifolds, In: Curvature and Topology of Riemannian Manifolds, Katata, 1985, (eds. K. Shiohama, T. Sakai and T. Sunada), Lecture Notes in Math., 1201 , Springer-Verlag, 1986, pp.,122-137.
[23] K. Matsuzaki, Isoperimetric constants for conservative Fuchsian groups, Kodai Math. J., 28 (2005), 292-300. · Zbl 1088.30041
[24] M. V. Melián, J. M. Rodríguez and E. Tourís, Escaping geodesics in Riemannian surfaces with pinched variable negative curvature, submitted.
[25] F. Paulin, On the critical exponent of a discrete group of hyperbolic isometries, Differential Geom. Appl., 7 (1997), 231-236. · Zbl 0885.53044
[26] D. Pestana, J. M. Rodríguez, J. M. Sigarreta and M. Villeta, Gromov hyperbolic cubic graphs, Cent. Eur. J. Math., 10 (2012), 1141-1151. · Zbl 1239.05141
[27] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Ann. of Math. Stud., 27 , Princeton University Press, 1951. · Zbl 0044.38301
[28] A. Portilla, J. M. Rodríguez and E. Tourís, Gromov hyperbolicity through decomposition of metrics spaces. II, J. Geom. Anal., 14 (2004), 123-149. · Zbl 1047.30028
[29] A. Portilla and E. Tourís, A characterization of Gromov hyperbolicity of surfaces with variable negative curvature, Publ. Mat., 53 (2009), 83-110. · Zbl 1153.53320
[30] B. Randol, Cylinders in Riemann surfaces, Comment. Math. Helv., 54 (1979), 1-5. · Zbl 0401.30036
[31] J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Grad. Texts in Math., 149 , Springer-Verlag, New York, 1994.
[32] J. M. Rodríguez, Isoperimetric inequalities and Dirichlet functions of Riemann surfaces, Publ. Mat., 38 (1994), 243-253. · Zbl 0813.30034
[33] J. M. Rodríguez, Two remarks on Riemann surfaces, Publ. Mat., 38 (1994), 463-477. · Zbl 0829.30027
[34] J. M. Rodríguez and E. Tourís, Gromov hyperbolicity through decomposition of metric spaces, Acta Math. Hungar., 103 (2004), 107-138. · Zbl 1051.30036
[35] J. M. Rodríguez and E. Tourís, Gromov hyperbolicity of Riemann surfaces, Acta Math. Sin. (Engl. Ser.), 23 (2007), 209-228.
[36] H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2), 77 (1963), 33-71. · Zbl 0218.10045
[37] P. M. Soardi, Rough isometries and Dirichlet finite harmonic functions on graphs, Proc. Amer. Math. Soc., 119 (1993), 1239-1248. · Zbl 0801.31002
[38] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25 (1987), 327-351. · Zbl 0615.53029
[39] E. Tourís, Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces, J. Math. Anal. Appl., 380 (2011), 865-881. · Zbl 1219.53047
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