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Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities

Published 26 Feb 2024 in math.DG and math.AP | (2402.16812v3)

Abstract: The ABP method for proving isoperimetric inequalities has been first employed by Cabr\'e in $\mathbb{R}n$, then developed by Brendle, notably in the context of non-compact Riemannian manifolds of non-negative Ricci curvature and positive asymptotic volume ratio. In this paper, we expand upon their approach and prove isoperimetric inequalities (sharp in the limit) in the presence of a small amount of negative curvature. First, we consider smallness of the negative part $\mathrm{Ric}-$ of the Ricci curvature in terms of its elliptic Kato constant. Indeed, the Kato constant turns out to control the non-negativity of the ($\infty$-)Bakry-\'Emery Ricci-tensor of a suitable conformal deformation of the manifold, and the ABP method can be implemented in this setting. Secondly, we show that the smallness of the Kato constant is ensured provided that the asymptotic volume ratio is positive and either $M$ has one end and asymptotically non-negative sectional curvature, or there is a suitable polynomial decay of $\mathrm{Ric}{-}$, and the relative volume comparison condition known as $\textbf{(VC)}$ holds. To show this latter fact, we enhance techniques elaborated by Li-Tam and Kasue to obtain new estimates of the Green function valid on the whole manifold.

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References (55)
  1. Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature. Invent. Math., 222(3):1033–1101, 2020.
  2. Mahaman Bazanfaré. A volume comparison theorem and number of ends for manifolds with asymptotically nonnegative Ricci curvature. Rev. Mat. Complut., 13(2):399–409, 2000.
  3. Mahaman Bazanfaré. Open manifolds with asymptotically nonnegative curvature. Illinois J. Math., 49(3):705–717, 2005.
  4. Arthur L. Besse. Einstein manifolds. Classics in Mathematics. Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition.
  5. Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature. Math. Ann., 385(3-4):1747–1773, 2023.
  6. Simon Brendle. Sobolev inequalities in manifolds with nonnegative curvature. Communications on Pure and Applied Mathematics.
  7. Simon Brendle. The isoperimetric inequality for a minimal submanifold in Euclidean space. J. Amer. Math. Soc., 34(2):595–603, 2021.
  8. Simon Brendle. The logarithmic Sobolev inequality for a submanifold in Euclidean space. Comm. Pure Appl. Math., 75(3):449–454, 2022.
  9. Xavier Cabre. Isoperimetric, sobolev, and eigenvalue inequalities via the alexandroff-bakelman-pucci method: a survey, 2015.
  10. Gilles Carron. Inégalités isopérimétriques sur les variétés riemanniennes. PhD thesis, Université Joseph Fourier, Grenoble, 1994.
  11. Gilles Carron. Geometric inequalities for manifolds with Ricci curvature in the Kato class. Ann. Inst. Fourier (Grenoble), 69(7):3095–3167, 2019.
  12. On four-dimensional steady gradient Ricci solitons that dimension reduce. Adv. Math., 403:Paper No. 108367, 61, 2022.
  13. Gaussian heat kernel estimates: from functions to forms. J. Reine Angew. Math., 761:25–79, 2020.
  14. The Poisson equation on Riemannian manifolds with weighted Poincaré inequality at infinity. Ann. Mat. Pura Appl. (4), 200(2):791–814, 2021.
  15. Limits of manifolds with a kato bound on the ricci curvature, 2021.
  16. Limits of manifolds with a kato bound on the ricci curvature. ii, 2022.
  17. Kato meets bakry-émery, 2023.
  18. Torus stability under Kato bounds on the Ricci curvature. J. Lond. Math. Soc. (2), 107(3):943–972, 2023.
  19. Geometric and spectral estimates based on spectral Ricci curvature assumptions. J. Reine Angew. Math., 772:121–145, 2021.
  20. Christopher B. Croke. A sharp four-dimensional isoperimetric inequality. Comment. Math. Helv., 59(2):187–192, 1984.
  21. Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana, 9(2):293–314, 1993.
  22. Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math., 28(3):333–354, 1975.
  23. Baptiste Devyver. Heat kernel and Riesz transform of Schrödinger operators. Ann. Inst. Fourier (Grenoble), 69(2):457–513, 2019.
  24. Baptiste Devyver. On gradient estimates for heat kernels. Comm. Partial Differential Equations, 46(5):717–779, 2021.
  25. Sobolev inequalities in manifolds with asymptotically nonnegative curvature, 2022.
  26. Sobolev inequality on manifolds with asymptotically nonnegative bakry-émery ricci curvature, 2022.
  27. The logarithmic Sobolev inequality for a submanifold in manifolds with asymptotically nonnegative sectional curvature. Acta Math. Sci. Ser. B (Engl. Ed.), 44(1):189–194, 2024.
  28. Xu Qian Fan. Poisson equation on some complete noncompact manifolds. Acta Math. Sin. (Engl. Ser.), 23(4):623–638, 2007.
  29. Normal and integral currents. Ann. of Math. (2), 72:458–520, 1960.
  30. Locally convex aspects of the kato and the dynkin class on manifolds, 2023.
  31. Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below. Math. Ann., 363(3-4):1307–1331, 2015.
  32. Alexander Grigor’yan. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.), 36(2):135–249, 1999.
  33. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
  34. Emmanuel Hebey. Nonlinear analysis on manifolds: Sobolev spaces and inequalities, volume 5 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.
  35. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math., 27:715–727, 1974.
  36. Florian Johne. Sobolev inequalities on manifolds with nonnegative Bakry-Émery ricci curvature, 2021.
  37. Atsushi Kasue. Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature. I. In Geometry and analysis on manifolds (Katata/Kyoto, 1987), volume 1339 of Lecture Notes in Math., pages 158–181. Springer, Berlin, 1988.
  38. Bruce Kleiner. An isoperimetric comparison theorem. Invent. Math., 108(1):37–47, 1992.
  39. Man-Chun Lee. Ricci Flow Under Kato-Type Curvature Lower Bound. J. Geom. Anal., 34(3):Paper no. 71, 2024.
  40. Peter Li. Geometric analysis, volume 134 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012.
  41. Symmetric Green’s functions on complete manifolds. Amer. J. Math., 109(6):1129–1154, 1987.
  42. Green’s functions, harmonic functions, and volume comparison. J. Differential Geom., 41(2):277–318, 1995.
  43. On the parabolic kernel of the Schrödinger operator. Acta Math., 156(3-4):153–201, 1986.
  44. Weighted Poincaré inequality and the Poisson equation. Trans. Amer. Math. Soc., 374(3):2167–2199, 2021.
  45. Sobolev inequalities in manifolds with nonnegative intermediate ricci curvature, 2023.
  46. Doanh Pham. Some Geometric Inequalities by the ABP Method. International Mathematics Research Notices, page rnad112, 06 2023.
  47. Christian Rose. Li-Yau gradient estimate for compact manifolds with negative part of Ricci curvature in the Kato class. Ann. Global Anal. Geom., 55(3):443–449, 2019.
  48. The Kato class on compact manifolds with integral bounds on the negative part of Ricci curvature. Proc. Amer. Math. Soc., 145(5):2199–2210, 2017.
  49. Eigenvalue estimates for Kato-type Ricci curvature conditions. Anal. PDE, 15(7):1703–1724, 2022.
  50. N. Th. Varopoulos. Hardy-Littlewood theory for semigroups. J. Funct. Anal., 63(2):240–260, 1985.
  51. André Weil. Sur les surfaces à courbure négative. C. R. Acad. Sci. Paris S´er. I Math., 182:4069–4071, 1926.
  52. Comparison geometry for the Bakry-Émery Ricci tensor. J. Differential Geom., 83(2):377–405, 2009.
  53. Yu Wang and Xiangwen Zhang. An Alexandroff-Bakelman-Pucci estimate on Riemannian manifolds. Adv. Math., 232:499–512, 2013.
  54. ABP estimate and geometric inequalities. Comm. Anal. Geom., 25(3):685–708, 2017.
  55. Qi S. Zhang and Z. Zhao. Estimates of global bounds for some Schrödinger heat kernels on manifolds. Illinois J. Math., 44(3):556–573, 2000.

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