[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/CKMS.c210097

NEW VOLUME COMPARISON WITH ALMOST RICCI SOLITON

azami, Shahroud (Department of Pure Mathematics Faculty of Science Imam Khomeini International University)
Hajiaghasi, Sakineh (Department of Pure Mathematics Faculty of Science Imam Khomeini International University)

Publication Information

Communications of the Korean Mathematical Society / v.37, no.3, 2022 , pp. 839-849 More about this Journal

Abstract

In this paper we consider a condition on the Ricci curvature involving vector fields which enabled us to achieve new results for volume comparison and Laplacian comparison. These results in special case obtained with considering volume non-collapsing condition. Also, by applying this condition we get new results of volume comparison for almost Ricci solitons.

Keywords

Volume comparison; Bakry-Emery Ricci curvature;

Citations & Related Records

Reference

1	J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119-128. http://projecteuclid.org/euclid.jdg/1214430220
2	X. Hu, D. Ji, and Y. Shi, Volume comparison of conformally compact manifolds with scalar curvature R ≥ -n(n - 1), Ann. Henri Poincare 17 (2016), no. 4, 953-977. https://doi.org/10.1007/s00023-015-0411-3 DOI
3	U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), no. 2, 355-374. https://doi.org/10.2307/1990957 DOI
4	J. Cheeger and T. H. Colding, Almost rigidity of warped products and the structure of spaces with Ricci curvature bounded below, C. R. Acad. Sci. Paris Ser. I Math. 320 (1995), no. 3, 353-357.
5	T. H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996), no. 1-3, 193-214. https://doi.org/10.1007/s002220050050 DOI
6	K. Grove and P. Petersen, Comparison Geometry, Mathematical Sciences Research Institute Publications, 30, Cambridge University Press, Cambridge, 1997.
7	J. Lott, Some geometric properties of the Bakry-Emery-Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865-883. https://doi.org/10.1007/s00014-003-0775-8 DOI
8	P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geom. Funct. Anal. 7 (1997), no. 6, 1031-1045. https://doi.org/10.1007/s000390050036 DOI
9	G. Wei, Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 311-313. https://doi.org/10.1090/S0273-0979-1988-15653-4 DOI
10	N. Yang, A note on nonnegative Bakry-Emery Ricci curvature, Arch. Math. (Basel) 93 (2009), no. 5, 491-496. https://doi.org/10.1007/s00013-009-0062-z DOI
11	S. Zhu, The comparison geometry of Ricci curvature, in Comparison geometry (Berkeley, CA, 1993-94), 221-262, Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997.
12	Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 190, 235-242. https://doi.org/10.1093/qmath/48.2.235 DOI
13	G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405. https://doi.org/10.4310/jdg/1261495336 DOI
14	W. Yuan, Volume comparison with respect to scalar curvature, Math. DG (2021).
15	Q. S. Zhang and M. Zhu, New volume comparison results and applications to degeneration of Riemannian metrics, Adv. Math. 352 (2019), 1096-1154. https://doi.org/10.1016/j.aim.2019.06.030 DOI