A necessary and sufficient condition for some steady Ricci solitons to have positive asymptotic volume ratio

Gao, Xiang

doi:10.2478/auom-2013-0027

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A necessary and sufficient condition for some steady Ricci solitons to have positive asymptotic volume ratio

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 21 (2013): Issue 2 (June 2013)

By: Xiang Gao

Open Access

|Sep 2013

Abstract

In this paper, we firstly establish a useful ODE relationship between R₁(c) and V₁(c) on the steady Ricci soliton. Based on this, we obtain a necessary and sufficient condition for some complete noncompact steady gradient Ricci solitons to have positive asymptotic volume ratio.

References

[1] H.-D. Cao, D. T. Zhou, On complete gradient shrinking Ricci solitons. J. Diff. Geom. 85, (2010), 175-185.10.4310/jdg/1287580963
Search in Google Scholar
[2] B.-L. Chen, Strong uniqueness of the Ricci flow. J. Diff. Geom. 82, (2009), 363-382.10.4310/jdg/1246888488
Search in Google Scholar
[3] B. Chow, P. Lu, B. Yang, A necessary and sufficient condition for Ricci shrinkers to have positive AVR. Proc. Amer. Math. Soc. 140 (2012), no. 6, 2179-2181.
Search in Google Scholar
[4] F.-Q. Fang, J.-W. Man, Z.-L. Zhang, Complete gradient shrinking Ricci solitons have finite topological type. C. R. Math. Acad. Sci. Paris 346, no. 11-12, (2008), 653-656.10.1016/j.crma.2008.03.021
Search in Google Scholar
[5] H. X. Guo, On the Ricci curvature of steady gradient Ricci solitons. J. Math. Anal. Appl. 363, (2010) 497-501.10.1016/j.jmaa.2009.09.046
Search in Google Scholar
[6] O. Munteanu, The volume growth of complete gradient shrinking Ricci solitons. arXiv: 0904. 0798.
Search in Google Scholar
[7] R. Schoen, S.-T. Yau, Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, (1994).
Search in Google Scholar
[8] S. J. Zhang, On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below. Acta Math. Sinica 27, (2011), 871-882. 10.1007/s10114-011-9527-7
Search in Google Scholar