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Isoperimetric inequalities under bounded integral norms of Ricci curvature and mean curvature

Author: Seong-Hun Paeng
Journal: Proc. Amer. Math. Soc. 146 (2018), 1309-1323
MSC (2010): Primary 53C20, 53C21
Published electronically: October 5, 2017
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Abstract: We obtain isoperimetric inequalities under bounded integral norms of Ricci curvature and mean curvature. Also we generalize the results to metric-measure spaces.

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  • [BE] D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177-206 (French). MR 889476,
  • [B] Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213-230. MR 683635
  • [Ch1] Isaac Chavel, Riemannian geometry--a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141
  • [Ch2] Isaac Chavel, Isoperimetric inequalities, Differential geometric and analytic perspectives, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001. MR 1849187
  • [DWZ] X. Dai, G. Wei, and Z. Zhang, Local Sobolev constant estimate for integral Ricci curvature bounds, arXiv:1601.0819 (2016).
  • [Ga] Sylvestre Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque 157-158 (1988), 191-216. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976219
  • [GW] Gregory J. Galloway and Eric Woolgar, Cosmological singularities in Bakry-Émery spacetimes, J. Geom. Phys. 86 (2014), 359-369. MR 3282334,
  • [Ha] Kit Hanes, Isoperimetric inequalities for manifolds with boundary, J. Differential Geometry 7 (1972), 525-534. MR 0334081
  • [HS] David Hoffman and Joel Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727. MR 0365424,
  • [Hs] Chuan-chih Hsiung, Isoperimetric inequalities for two-dimensional Riemannian manifolds with boundary, Ann. of Math. (2) 73 (1961), 213-220. MR 0130637,
  • [L] John Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865-883. MR 2016700,
  • [Pa1] Seong-Hun Paeng, Buser's isoperimetric inequalities with integral norms of Ricci curvature, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2903-2910. MR 2801631,
  • [Pa2] Seong-Hun Paeng, Volumes and intrinsic diameters of hypersurfaces, J. Geom. Phys. 95 (2015), 96-107. MR 3357824,
  • [PW] P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1031-1045. MR 1487753,
  • [R] William T. Reid, The isoperimetric inequality and associated boundary problems, J. Math. Mech. 8 (1959), 897-905. MR 0130623
  • [WW] Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405. MR 2577473

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Additional Information

Seong-Hun Paeng
Affiliation: Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seoul 143-701, Republic of Korea

Keywords: Isoperimetric inequality, Ricci curvature, mean curvature
Received by editor(s): November 23, 2016
Received by editor(s) in revised form: May 10, 2017
Published electronically: October 5, 2017
Additional Notes: This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03931459).
Communicated by: Guofang Wei
Article copyright: © Copyright 2017 American Mathematical Society

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