Neumann isoperimetric constant estimate for convex domains

Dai, Xianzhe; Wei, Guofang; Zhang, Zhenlei

doi:10.1090/proc/14079

Neumann isoperimetric constant estimate for convex domains
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by Xianzhe Dai, Guofang Wei and Zhenlei Zhang PDF

Proc. Amer. Math. Soc. 146 (2018), 3509-3514 Request permission

Abstract:

We present a geometric and elementary proof of the local Neumann isoperimetric inequality on convex domains of a Riemannian manifold with Ricci curvature bounded below.

References

M. T. Anderson, The $L^2$ structure of moduli spaces of Einstein metrics on $4$-manifolds, Geom. Funct. Anal. 2 (1992), no. 1, 29–89. MR 1143663, DOI 10.1007/BF01895705
Vincent Bayle and César Rosales, Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds, Indiana Univ. Math. J. 54 (2005), no. 5, 1371–1394. MR 2177105, DOI 10.1512/iumj.2005.54.2575
Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
Fabio Cavalletti and Andrea Mondino, Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds, Invent. Math. 208 (2017), no. 3, 803–849. MR 3648975, DOI 10.1007/s00222-016-0700-6
Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
Jeff Cheeger and Shing Tung Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465–480. MR 615626, DOI 10.1002/cpa.3160340404
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
Xianzhe Dai, Guofang Wei, and Zhenlei Zhang, Local Sobolev constant estimate for integral Ricci curvature bounds, Adv. Math. 325 (2018), 1–33. MR 3742584, DOI 10.1016/j.aim.2017.11.024
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
Sylvestre Gallot, Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, Astérisque 163-164 (1988), 5–6, 31–91, 281 (1989) (French, with English summary). On the geometry of differentiable manifolds (Rome, 1986). MR 999971
M. Gromov, Paul Levy’s isoperimetric inequality, Publications IHES, 1980.
Peter Li, Geometric analysis, Cambridge Studies in Advanced Mathematics, vol. 134, Cambridge University Press, Cambridge, 2012. MR 2962229, DOI 10.1017/CBO9781139105798
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
Fanghua Lin and Xiaoping Yang, Geometric measure theory—an introduction, Advanced Mathematics (Beijing/Boston), vol. 1, Science Press Beijing, Beijing; International Press, Boston, MA, 2002. MR 2030862
Emanuel Milman, Sharp isoperimetric inequalities and model spaces for the curvature-dimension-diameter condition, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041–1078. MR 3346688, DOI 10.4171/JEMS/526
Frank Morgan, The Levy-Gromov isoperimetric inequality in convex manifolds with boundary, J. Geom. Anal. 18 (2008), no. 4, 1053–1057. MR 2438911, DOI 10.1007/s12220-008-9047-6
Lei Ni and Kui Wang, Isoperimetric comparisons via viscosity, J. Geom. Anal. 26 (2016), no. 4, 2831–2841. MR 3544942, DOI 10.1007/s12220-015-9650-2
Laurent Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), no. 2, 417–450. MR 1180389
Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. MR 2577473, DOI 10.4310/jdg/1261495336
Deane Yang, Convergence of Riemannian manifolds with integral bounds on curvature. I, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 1, 77–105. MR 1152614
Z. Zhang, Notes on the isoperimetric constant estimate, (2016), preprint.