Abstract
We prove that an n-dimensional, \(n\ge 4\), compact gradient shrinking Ricci soliton satisfying a \(L^{\frac{n}{2}}\)-pinching condition is isometric to a quotient of the round \(\mathbb {S}^n\), which improves the rigidity theorem given by Catino (Integral pinched shrinking Ricci solitons, 2016), in dimension \(4\le n\le 6\).
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References
Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998)
Cao, H. D.: Recent progress on Ricci solitons in recent advances in geometric analysis. In: Advanced Lectures in Mathematics (ALM), vol. 11, pp.1–38. International Press, Somerville, MA, (2010)
Cation, G.: Integral pinched shrinking Ricci solitons. Adv. Math. 303, 279–294 (2016)
Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons: the equation point of view. Manuscr. Math. 127, 345–367 (2008)
Fu, H.P., Xiao, L.Q.: Einstein manifolds with finite \(L^p\)-norm of the Weyl curvature. Submitted to Differ. Geom. Appl. in September 1st (2015)
Hamilton, R.S.: The Ricci flow on surfaces. Mathematics and general relativity, (Santa Cruz, CA, 1986), pages 237-262. Contemp. Math. 71, 237–262 (1988). (Am. Math. Soc.)
Hebey, E., Vaugon, M.: Effective \(L^p\) pinching for the concircular curvature. J. Geom. Anal. 6, 531–553 (1996)
Huisken, G.: Ricci deformation of the metric on a Riemannian manifold. J. Differ. Geom. 21, 47–62 (1985)
Itoh, M., Satoh, H.: Isolation of the Weyl conformal tensor for Einstein manifolds. Proc. Jpn. Acad. A 78, 140–142 (2002)
Ivey, T.: Ricci solitons on compact three-manifolds. Differ. Geom. Appl. 3, 301–307 (1993)
Lee, M.J., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987)
Munteanu, O., Wang, J.: Positively curved shrinking Ricci solitons are compact. arXiv: 1504.07898vl [math.DG], (2015)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159vl [math.DG], (2002)
Singer, M.: Positive Einstein metrics with small \(L^{n/2}\)-norm of the Weyl tensor. Differ. Geom. Appl. 2, 269–274 (1992)
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The authors would like to thank the referee for some helpful suggestions.
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Communicated by A. Constantin.
Supported by National Natural Science Foundations of China (11261038, 11361041), Jiangxi Province Natural Science Foundation of China (20132BAB201005).
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Fu, HP., Xiao, LQ. Rigidity theorem for integral pinched shrinking Ricci solitons. Monatsh Math 183, 487–494 (2017). https://doi.org/10.1007/s00605-017-1042-1
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DOI: https://doi.org/10.1007/s00605-017-1042-1