Comparison geometry for integral Bakry-Émery Ricci tensor bounds. (English) Zbl 1408.53048
J. Geom. Anal. 29, No. 1, 828-867 (2019); correction ibid 30, No. 4, 4464-4465 (2020).
Summary: We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie’s comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates, and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry-Émery Ricci tensor. These give generalizations of some work of Petersen-Wei, Aubry, Petersen-Sprouse, Yau and more.
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
Keywords:
Bakry-Émery Ricci tensor; smooth metric measure space; integral curvature; comparison theorem; diameter estimate; eigenvalue estimate; volume growth estimateReferences:
| [1] | Aubry, E.: Finiteness of \[\pi_1\] π1 and geometric inequalities in almost positive Ricci curvature. Ann. Sci. Ecole Norm. Sup. 40, 675-695 (2007) · Zbl 1141.53034 · doi:10.1016/j.ansens.2007.07.001 |
| [2] | Aubry, E.: Bounds on the volume entropy and simplicial volume in Ricci curvature \[L^p\] Lp-bounded from below. Int. Math. Res. Not. IMRN 10, 1933-1946 (2009) · Zbl 1182.53034 |
| [3] | Bakry, D., Émery, M.: Diffusion hypercontractivitives. In: Séminaire de Probabilités XIX, 1983/1984. Lecture Notes in Math., vol. 1123, pp. 177-206. Springer, Berlin (1985) · Zbl 0561.60080 |
| [4] | Bakry, D., Qian, Z.-M.: Some new results on eigenvectors via dimension, diameter and Ricci curvature. Adv. Math. 155, 98-153 (2000) · Zbl 0980.58020 · doi:10.1006/aima.2000.1932 |
| [5] | Bakry, D., Qian, Z.-M.: Volume comparison theorems without Jacobi fields. In: Current Trends in Potential Theory, Theta Ser. Adv. Math., vol. 4, pp. 115-122 (2005) · Zbl 1212.58019 |
| [6] | Cheng, S.-Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143, 289-297 (1975) · Zbl 0329.53035 · doi:10.1007/BF01214381 |
| [7] | Colding, T.H.: Ricci curvature and volume convergence. Ann. Math. (2) 145, 477-501 (1997) · Zbl 0879.53030 · doi:10.2307/2951841 |
| [8] | Dai, X.-Z., Petersen, P., Wei, G.-F.: Integral pinching theorems. Manuscr. Math. 101, 143-152 (2000) · Zbl 0970.53022 · doi:10.1007/s002290050009 |
| [9] | Dai, X.-Z., Wei, G.-F.: A heat kernel lower bound for integral Ricci curvature. Mich. Math. J. 52, 61-69 (2004) · Zbl 1086.53059 · doi:10.1307/mmj/1080837734 |
| [10] | Dai, X.-Z., Wei, G.-F., Zhang, Z.-L.: Local sobolev constant estimate for integral Ricci curvature bounds. arXiv:1601.08191 · Zbl 1380.53041 |
| [11] | Gallot, S.: Isoperimetric inequalities based on integral norms of Ricci curvature. Astérisque. 157-158, 191-216, (1988). Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987) · Zbl 0665.53041 |
| [12] | Hamilton, R.: The formation of singularities in the Ricci flow. Surv. Differ. Geom. 2, 7-136 (1995) · Zbl 0867.53030 |
| [13] | Hu, Z.-S., Xu, S.-L.: Bounds on the fundamental groups with integral curvature bound. Geom. Dedicata 143, 1-16 (2008) · Zbl 1143.53034 · doi:10.1007/s10711-008-9235-3 |
| [14] | Li, X.-D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pure. Appl. 84, 1295-1361 (2005) · Zbl 1082.58036 · doi:10.1016/j.matpur.2005.04.002 |
| [15] | Limoncu, M.: The Bakry-Emery Ricci tensor and its applications to some compactness theorems. Math. Z. 271, 715-722 (2012) · Zbl 1264.53042 · doi:10.1007/s00209-011-0886-7 |
| [16] | Lott, J.: Some geometric properties of the Bakry-Émery-Ricci tensor. Comment. Math. Helv. 78, 865-883 (2003) · Zbl 1038.53041 · doi:10.1007/s00014-003-0775-8 |
| [17] | Petersen, P., Sprouse, C.: Integral curvature bounds, distance estimates and applications. J. Differ. Geom. 50, 269-298 (1998) · Zbl 0969.53017 · doi:10.4310/jdg/1214461171 |
| [18] | Petersen, P., Wei, G.-F.: Relative volume comparison with integral curvature bounds. GAFA 7, 1031-1045 (1997) · Zbl 0910.53029 |
| [19] | Petersen, P., Wei, G.-F.: Analysis and geometry on manifolds with integral Ricci curvature bounds. II. Trans. AMS. 353, 457-478 (2000) · Zbl 0999.53030 · doi:10.1090/S0002-9947-00-02621-0 |
| [20] | Tadano, H.: Remark on a diameter bound for complete Riemannian manifolds with positive Bakry-Émery Ricci curvature. Differ. Geom. Appl. 44, 136-143 (2016) · Zbl 1330.53051 · doi:10.1016/j.difgeo.2015.11.001 |
| [21] | Wei, G.-F., Wylie, W.: Comparison geometry for the smooth metric measure spaces. In: Proceedings of the 4th International Congress of Chinese Mathematicians, vol. 2, pp. 191-202 (2007) |
| [22] | Wei, G.-F., Wylie, W.: Comparison geometry for the Bakry-Émery Ricci tensor. J. Differ. Geom. 83, 377-405 (2009) · Zbl 1189.53036 · doi:10.4310/jdg/1261495336 |
| [23] | Yang, D.: Convergence of Riemannian manifolds with integral bounds on curvature I. Ann. Sci. École Norm. Sup. 25, 77-105 (1992) · Zbl 0748.53025 · doi:10.24033/asens.1644 |
| [24] | Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25, 659-670 (1976) · Zbl 0335.53041 · doi:10.1512/iumj.1976.25.25051 |
| [25] | Zhu, S.-H.: The comparison geometry of Ricci curvature. In: Comparison geometry (Berkeley, CA, 1993-94), Math. Sci. Res. Inst. Publ, vol. 30, pp. 221-262. Cambridge University Press, Cambridge (1997) · Zbl 0896.53036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
