A gap theorem on complete shrinking gradient Ricci solitons
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Abstract:
In this short note, using Günther’s volume comparison theorem and Yokota’s gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature $K(g)<A$ and $\textrm {Vol}_{f}(M)\geq v$ for some uniform constant $A,v$, there exists a small uniform constant $\epsilon _{n,A,v}>0$ depends only on $n, A$ and $v$, if the scalar curvature $R\leq \epsilon _{n,A,v}$, then $(M,g,f)$ is isometric to the Gaussian soliton $(\mathbb {R}^{n}, g_{E}, \frac {|x|^{2}}{4})$.References
- José A. Carrillo and Lei Ni, Sharp logarithmic Sobolev inequalities on gradient solitons and applications, Comm. Anal. Geom. 17 (2009), no. 4, 721–753. MR 3010626, DOI 10.4310/CAG.2009.v17.n4.a7
- Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. MR 2732975
- Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
- Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. MR 2520796
- Chih-Wei Chen, On the asymptotic behavior of expanding gradient Ricci solitons, Ann. Global Anal. Geom. 42 (2012), no. 2, 267–277. MR 2947955, DOI 10.1007/s10455-012-9311-7
- H. D. Cao, R. Hamilton, and T. Ilmanen, Gaussian densities and stability for some Ricci solitons, arXiv: math/0404165.
- Bennett Chow, Peng Lu, and Bo Yang, Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons, C. R. Math. Acad. Sci. Paris 349 (2011), no. 23-24, 1265–1267 (English, with English and French summaries). MR 2861997, DOI 10.1016/j.crma.2011.11.004
- Bennett Chow, Peng Lu, and Bo Yang, A necessary and sufficient condition for Ricci shrinkers to have positive AVR, Proc. Amer. Math. Soc. 140 (2012), no. 6, 2179–2181. MR 2888203, DOI 10.1090/S0002-9939-2011-11173-0
- Fu-quan Fang, Jian-wen Man, and Zhen-lei Zhang, Complete gradient shrinking Ricci solitons have finite topological type, C. R. Math. Acad. Sci. Paris 346 (2008), no. 11-12, 653–656 (English, with English and French summaries). MR 2423272, DOI 10.1016/j.crma.2008.03.021
- H. Ge and S. Zhang, Liouville type theorems on complete shrinking gradient Ricci solitons, arXiv:1609.09806.
- Paul Günther, Einige Sätze über das Volumenelement eines Riemannschen Raumes, Publ. Math. Debrecen 7 (1960), 78–93 (German). MR 141058
- Robert Haslhofer and Reto Müller, A compactness theorem for complete Ricci shrinkers, Geom. Funct. Anal. 21 (2011), no. 5, 1091–1116. MR 2846384, DOI 10.1007/s00039-011-0137-4
- Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. MR 2460872, DOI 10.2140/gt.2008.12.2587
- Ovidiu Munteanu and Mu-Tao Wang, The curvature of gradient Ricci solitons, Math. Res. Lett. 18 (2011), no. 6, 1051–1069. MR 2915467, DOI 10.4310/MRL.2011.v18.n6.a2
- Aaron Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645 (2010), 125–153. MR 2673425, DOI 10.1515/CRELLE.2010.062
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
- Stefano Pigola, Michele Rimoldi, and Alberto G. Setti, Remarks on non-compact gradient Ricci solitons, Math. Z. 268 (2011), no. 3-4, 777–790. MR 2818729, DOI 10.1007/s00209-010-0695-4
- Peter Topping, Diameter control under Ricci flow, Comm. Anal. Geom. 13 (2005), no. 5, 1039–1055. MR 2216151
- Takumi Yokota, Perelman’s reduced volume and a gap theorem for the Ricci flow, Comm. Anal. Geom. 17 (2009), no. 2, 227–263. MR 2520908, DOI 10.4310/CAG.2009.v17.n2.a3
- Takumi Yokota, Addendum to ‘Perelman’s reduced volume and a gap theorem for the Ricci flow’ [MR2520908], Comm. Anal. Geom. 20 (2012), no. 5, 949–955. MR 3053617, DOI 10.4310/CAG.2012.v20.n5.a2
- Shi Jin 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. MR 2786449, DOI 10.1007/s10114-011-9527-7
Additional Information
- Shijin Zhang
- Affiliation: School of Mathematics and systems science, Beihang University, Beijing, 100871, People’s Republic of China
- MR Author ID: 887805
- Email: shijinzhang@buaa.edu.cn
- Received by editor(s): December 30, 2016
- Received by editor(s) in revised form: February 5, 2017, and February 6, 2017
- Published electronically: August 7, 2017
- Communicated by: Guofang Wei
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 359-368
- MSC (2010): Primary 53C20
- DOI: https://doi.org/10.1090/proc/13689
- MathSciNet review: 3723146