Abstract
In this paper, we study the evolving behaviors of the first eigenvalue of the Laplace–Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.
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Cao, X. D.: Eigenvalues of (\(( - \vartriangle + \frac{R}{2})\)) on manifolds with nonnegative curvature operator. Math. Ann., 337(2), 435–441 (2007)
Cao, X. D.: First eigenvalues of geometric operators under the Ricci flow. Proc. Amer. Math. Soc., 136(11), 4075–4078 (2008)
Cao, X. D., Guckenheimer, J., Saloff-Coste, L.: The backward behavior of the Ricci and cross curvature flows on SL(2,ℝ). Comm. Anal. Geom., 17(4), 777–796 (2009)
Cao, X. D., Hou, S. B., Ling J.: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann., 354, 451–463 (2012)
Cao, X. D., Saloff-Coste, L.: Backward Ricci flow on locally homogeneous three-manifolds. Comm. Anal. Geom., 12(2), 305–325 (2009)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, Grad. Stud. Math., vol. 77, American Mathematical Society/Science Press, Providence, RI/New York, 2006
Hamilton, R. S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom., 17(2), 255–306 (1982)
Hou, S. B.: Eigenvalues under the Ricci flow of model geometries. Acta Math. Sin., Chinese Ser., 60(4), 583–594 (2017)
Isenberg, J., Jacken, M.: Ricci flow of locally homogeneous geometries of closed manifolds. J. Differ. Geome., 35 723–741 (1992)
Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol., 12(5), 2587–2855 (2008)
Knopf, D., McLeod, K.: Quasi-convergence of model geometries under the Ricci flow. Comm. Anal. Geom., 9(4), 879–919 (2001)
Li, J. F.: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann., 338, 927–946 (2007)
Ling, J.: A class of monotonic quantities along the Ricci flow. arXiv:0710.4291v2, 2007
Ma, L.: Eigenvalue monotonicity for the Ricci–Hamilton flow. Ann. Glob. Anal. Geom., 29(3), 287–292 (2006)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1, 2002
Singer, I. M.: Infinitesimally homogeneous spaces. Comm. Pure Appl. Math., 13, 685–697 (1960)
Wu, J. Y.: First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow. Acta Math. Sin., Engl. Ser., 27(8), 1591–1598 (2011)
Wu, J. Y., Wang, E., Zheng, Y.: First eigenvalue of the p-Laplace operator along the Ricci flow. Ann. Glob. Anal. Geom., 38(1), 27–55 (2010)
Acknowledgements
The author would like to thank Professor Xiaodong Cao and Professor Laurent Saloff-Coste for their suggestions and interests in this work. The author would also like to thank referees for their valuable comments.
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Supported by NSFC (Grant No. 11001268) and Chinese Universities Scientific Fund (Grant No. 2014QJ002)
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Hou, S.B. Eigenvalues under the Backward Ricci Flow on Locally Homogeneous Closed 3-manifolds. Acta. Math. Sin.-English Ser. 34, 1179–1194 (2018). https://doi.org/10.1007/s10114-018-6448-8
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DOI: https://doi.org/10.1007/s10114-018-6448-8