Abstract
Let (M, g) be a compact Riemannian manifold without boundary or with smooth boundary having vanishing mean curvature, and \(g=g(t)\) are one-parameter family of Riemannian metrics evolving by the Yamabe flow. In this paper, we derive some evolution equations for the first eigenvalue of the operators \(-\Delta +aR^\alpha \), \(0<\alpha \le 1\) for some positive constant a under the (unnormalized and normalized) Yamabe flow. Some monotonic quantities depending on the first eigenvalue are obtained under certain restrictions on the constant a as an application of the evolution equations.
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References
Abolarinwa, A.: Eigenvalues of weighted Laplacian under the extended Ricci flow. Adv. Geom. 19(1), 131–143 (2019)
Abolarinwa, A.: Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow. J. Appl. Anal. 21(2), 147–160 (2015)
Aubin, T.: Équations différentielles non linéaires et probleme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)
Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin (1998)
Azami, S.: First eigenvalues of geometric operator under the Ricci-Bourguignon flow. J. Indones. Math. Soc. 24(1), 51–60 (2018)
Azami, S.: Evolution of eigenvalues of geometric operator under the rescaled List’s extended Ricci flow. Bull. Iran. Math. Soc. 48, 1265–1279 (2020)
Azami, S., Abolarinwa, A.: Evolution of Yamabe constant along the Ricci-Bourguignon flow. Arab J. Math. (2022). https://doi.org/10.1007/s40065-022-00376-y
Barbosa, E., Ribeiro, E., Jr.: On conformal solutions of the Yamabe flow. Arch. Math. (Basel) 101, 79–89 (2013)
Brendle, S.: A generalization of the Yamabe flow for manifolds with boundary. Asian J. Math. 6, 625–644 (2002)
Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69, 217–278 (2005)
Cao, X.D.: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc. 136, 4075–4078 (2008)
Cao, X.D.: Eigenvalues of\((-\Delta +\frac{R}{2})\)on manifols with nonnegative curvature operator. Math. Ann. 337(2), 435–441 (2007)
Cao, X., Hou, S., Ling, J.: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354, 451–463 (2012)
Chow, B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Commun. Pur Appl. Math. 45, 1003–1014 (1992)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, Graduate Studies in Mathematics, vol. 77. American Mathematical Society, Providence (2006)
Escobar, J.F.: The Yamabe problem on manifolds with boundary. J. Differ. Geom. 35, 21–84 (1992)
Fang, S., Yang, F.: First eigenvalues of geometric operators under the Yamabe flow. Bull. Korean Math. Soc. 53, 1113–1122 (2016)
Ho, P.T.: First eigenvalues of geometric operators under the Yamabe flow. Ann. Glob. Anal. Geom. 54, 449–472 (2018)
Hamilton, R. S.: Lectures on geometric flows, unpublished (1989)
Kato, T.: Perturbation Theory for Linear Operator, 2nd edn Springer, Berlin (1984)
Kleiner, B., Lott, J.: Note on Perelman’s papers. Geom. Topol. 12, 2587–2855 (2008)
Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987)
Li, Y., Abolarinwa, A., Azami, S., Ali, A.: Yamabe constant evolution and monotonicity along the conformal Ricci flow. AIMS Math. 7(7), 12077–12090 (2022)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, Ar**v math. ar**v:math/0211159 (2002)
Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)
Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for “large’’ energies. J. Reine Angew. Math. 562, 59–100 (2003)
Suárez-Serrato, P., Tapie, S.: Conformal entropy rigidity through Yamabe flows. Math. Ann. 353, 333–357 (2012)
Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 22, 265–274 (1968)
Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)
Ye, R.: Global existence and convergence of Yamabe flow. J. Differ. Geom. 39, 35–50 (1994)
Yang, F., Zhang, L.: On the evolution and monotonicity of first eigenvalues under the Ricci flow. Hokkaido Math. J. 51, 107–116 (2022)
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Abolarinwa, A., Azami, S. First eigenvalues evolution for some geometric operators along the Yamabe flow. J. Geom. 115, 18 (2024). https://doi.org/10.1007/s00022-024-00717-6
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DOI: https://doi.org/10.1007/s00022-024-00717-6