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http://dx.doi.org/10.4134/BKMS.b170306

ON A CLASS OF COMPLETE NON-COMPACT GRADIENT YAMABE SOLITONS  

Wu, Jia-Yong (Department of Mathematics Shanghai Maritime University)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.3, 2018 , pp. 851-863 More about this Journal
Abstract
We derive lower bounds of the scalar curvature on complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that potential functions of Yamabe solitons have at most quadratic growth for distance function. We also obtain a finite topological type property on complete shrinking gradient Yamabe solitons under suitable scalar curvature assumptions.
Keywords
Yamabe flow; Yamabe soliton; self-similar solution; finite topological type;
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1 R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1986.
2 P. Li and S.-T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986), no. 3-4, 153-201.   DOI
3 L. Ma and L. Cheng, Properties of complete non-compact Yamabe solitons, Ann. Global Anal. Geom. 40 (2011), no. 3, 379-387.   DOI
4 L. Ma and V. Miquel, Remarks on scalar curvature of Yamabe solitons, Ann. Global Anal. Geom. 42 (2012), no. 2, 195-205.   DOI
5 S. J. Zhang, On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 5, 871-882.
6 Z.-H. Zhang, On the completeness of gradient Ricci solitons, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2755-2759.   DOI
7 H.-D. Cao, Geometry of complete gradient shrinking Ricci solitons, in Geometry and analysis. No. 1, 227-246, Adv. Lect. Math. (ALM), 17, Int. Press, Somerville, MA, 2011.
8 E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45-56.   DOI
9 H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 121-142.   DOI
10 H.-D. Cao, Recent progress on Ricci solitons, in Recent advances in geometric analysis, 1-38, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010.
11 H.-D. Cao, X. Sun, and Y. Zhang, On the structure of gradient Yamabe solitons, Math. Res. Lett. 19 (2012), no. 4, 767-774.   DOI
12 H.-D. Cao and D. Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175-185.   DOI
13 G. Catino, C. Mantegazza, and L. Mazzieri, On the global structure of conformal gradient solitons with nonnegative Ricci tensor, Commun. Contemp. Math. 14 (2012), no. 6, 1250045, 12 pp.
14 B.-L. Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363-382.   DOI
15 P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math. 240 (2013), 346-369.   DOI
16 L. F. Di Cerbo and M. M. Disconzi, Yamabe solitons, determinant of the Laplacian and the uniformization theorem for Riemann surfaces, Lett. Math. Phys. 83 (2008), no. 1, 13-18.   DOI
17 F. Fang, J. Man, and Z. Zhang, Complete gradient shrinking Ricci solitons have finite topological type, C. R. Math. Acad. Sci. Paris 346 (2008), no. 11-12, 653-656.   DOI
18 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.   DOI
19 B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (1992), no. 8, 1003-1014.   DOI
20 B. Chow, P. Lu, and L. Ni, Hamilton's Ricci flow, Lectures in Contemporary Mathe-matics 3, Science Press and American Mathematical Society, 2006.