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Evolution of Eigenvalues along Rescaled Ricci Flow

Published online by Cambridge University Press:  20 November 2018

Junfang Li
Junfang Li*
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294 e-mail: jli@math.uab.edu
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Abstract

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In this paper, we discuss monotonicity formulae of various entropy functionals under various rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue of a family of geometric operators $-4\Delta \,+\,kR$ is monotonic along the normalized Ricci flow for all $k\,\ge \,1$ provided the initial manifold has nonpositive total scalar curvature.

Keywords

58C4053C44monotonicity formulasRicci flow
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 127 - 135
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Akutagawa, K., Ishida, M., and LeBrun, C., Perelman's invariant, Ricci flow, and the Yamabe invariants of smooth manifolds. Arch. Math. 88 (2007), no. 1, 71–76.Google Scholar
[2] Cao, X., Eigenvalues of(-Δ R/2 ) on manifolds with nonnegative curvature operator. Math. Ann. 337 (2007), no. 2, 435–441.Google Scholar
[3] Cao, X., First eigenvalues of geometric operators under Ricci flow. arxiv:math.DG/0710.3947v1.Google Scholar
[4] Chow, B. and Knopf, D., The Ricci Flow: An Introduction. Mathematical Surveys and Monographs 110. American Mathematical Society, Providence, RI, 2004.Google Scholar
[5] Chow, B., S.-C. Chu, Glickenstein, D., et al., The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects. Mathematical Surveys and Monographs 135. American Mathematical Society, Providence, RI, 2007.Google Scholar
[6] Hamilton, R. S., The formation of singularities in the Ricci flow. In: Surveys in Differential Geometry, Vol. II. Internat. Press, Cambridge, MA, 1995, pp. 7–136.Google Scholar
[7] Ivey, T., Ricci solitons on compact three-manifolds. Differential Geom. Appl. 3 (1993), no. 4, 301–307. http://dx.doi.org/10.1016/0926-2245(93)90008-O Google Scholar
[8] Li, J.-F., Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338 (2007), 927–946. http://dx.doi.org/10.1007/s00208-007-0098-y Google Scholar
[9] Oliynyka, T., Suneetab, V., and E.Woolgara, Irreversibility of world-sheet renormalization group flow. Phys. Lett. B,610 (2005), no. 1-2, 115–121. http://dx.doi.org/10.1016/j.physletb.2005.01.077 Google Scholar
[10] Perelman, G., The entropy formula for the Ricci flow and its geometric applications. arxiv:math.DG/0211159.Google Scholar