Geodesic
The Isoperimetric Inequality on Manifolds of Conformally Hyperbolic Type
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 2, pp. 12-23 
V. A. Zorich; V. M. Kesel'man. The Isoperimetric Inequality on Manifolds of Conformally Hyperbolic Type. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 2, pp. 12-23. http://geodesic.mathdoc.fr/item/FAA_2001_35_2_a1/
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     title = {The {Isoperimetric} {Inequality} on {Manifolds} of {Conformally} {Hyperbolic} {Type}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that the maximal isoperimetric function on a Riemannian manifold of conformally hyperbolic type can be reduced to the linear canonical form $P(x)=x$ by a conformal change of the Riemannian metric. In other words, the isoperimetric inequality $P(V(D))\le S(\partial D)$, relating the volume $V(D)$ of a domain $D$ to the area $S(\partial D)$ of its boundary, can be reduced to the form $V(D)\le S(\partial D)$, known for the Lobachevskii hyperbolic space.

[1] Zorich V. A., Keselman V. M., “O konformnom tipe rimanova mnogoobraziya”, Funkts. analiz i ego pril., 30:2 (1996), 40–55 | DOI | MR | Zbl

[2] Zorich V. A., “Asymptotic geometry and conformal types of Carnot–Carathéodory spaces”, Geom. Funct. Anal., 9:2 (1999), 393–411 | DOI | MR | Zbl

[3] Grimaldi R., Pansu P., “Sur la croissance du volume dans une classe conforme”, J. Math. Pures Appl., 9(71):1 (1992), 1–19 | MR

[4] Zorich V. A., Keselman V. M., “Kanonicheskii vid izoperimetricheskogo neravenstva na mnogoobraziyakh konformno-giperbolicheskogo tipa”, UMN, 54:3 (1999), 165–166 | DOI | MR | Zbl

[5] Gromov M., Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, Boston–Basel–Berlin, 1999, with appendices by Katz M., Pansu P., and Semmes S. | MR | Zbl

[6] Zorich V. A., Keselman V. M., “Konformnyi tip i izoperimetricheskaya razmernost rimanova mnogoobraziya”, Matem. zametki, 63:3 (1998), 379–385 | DOI | MR | Zbl

[7] Mazya V. G., Prostranstva S. L. Soboleva, Izd-vo Leningradskogo un-ta, L., 1985 | MR

[8] Reshetnyak Yu. G., Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk, 1982 | MR

[9] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[10] Heinonen J., Kilpelainen T., Martio O., Nonlinear potential theory of degenerate elliptic equations, Oxford University Press, 1993 | MR | Zbl

[11] Burago Yu. D., Zalgaller V. A., Geometricheskie neravenstva, Nauka, L., 1980 | MR | Zbl

[12] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987 | MR | Zbl

[13] Ahlfors L., “Sur le type d'une surface de Riemann”, C. R. Acad. Sci. Paris, Ser. A, 201 (1935), 30–32 | MR | Zbl

[14] Ahlfors L., Calabi E., Morse M., Sario L., Spencer D. (eds.), Contributions to the theory of Riemann surfaces, Centennial celebration of Riemann's dissertation, Ann. of Math. Stud., 30, Princeton University Press, Princeton, 1953 | MR | Zbl

[15] Cheeger J., “A lower bound for the smallest eigenvalue of the Laplacian”, Problems in Analysis: A Symposium in Honor of Salomon Bochner, Princeton University Press, Princeton, 1970, 195–199 | MR | Zbl

[16] Cheng S. V., Yau S. T., “Differential equations on Riemannian manifolds and their geometric applications”, Comm. Pure Appl. Math., 28 (1975), 333–354 | DOI | MR | Zbl

[17] Milnor J., “On deciding whether a surface is parabolic or hyperbolic”, Amer. Math. Monthly, 84 (1977), 43–46 | DOI | MR | Zbl

[18] Varopoulos N., Saloff-Coste L., Coulhon T., Analysis and geometry on groups, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[19] Grigor'yan A., “Analytic and geometric background of recurrence and non-explosion of the Brownian motion”, Bull. Amer. Math. Soc. (N.S.), 36:2 (1999), 135–249 | DOI | MR

[20] Grigor'yan A., “Isoperimetric inequality and capacities on Riemannian manifolds”, Oper. Theory Adv. Appl., 109, Birkhäuser Verlag, Basel, 1999, 139–153 | MR