First Eigenvalue of p-Laplacian Along The Normalized Ricci Flow on Bianchi Classes

Mohammad Javad Habibi Vosta Kolaei, Shahroud Azami

Abstract


Consider M as a 3-homogeneous manifold. In this paper, we are going to study the behavior of the first eigenvalue of p-Laplace operator in a case of Bianchi classes along the normalized Ricci flow also we will give some upper and lower bounds for a such eigenvalue.

Keywords


Ricci flow, p-Laplacian operator, Eigenvalue.

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References


A. Abolarinwa, Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci harmonic flow, J. Appl. Anal., 21(2) (2015), 147-160.

S. Azami, Eigenvalue variation of the p-Laplacian under the Yamabe flow, Cogent Mathematics, 3 (2016), 1236566.

S. Azami, Monotonicity of eigenvalues of Witten-Laplace operator along the RicciBourguignon flow, AIMS mathematics, 2(2)(2017), 230-243.

X. Cao, Eigenvalues of −∆ + R2 on manifolds with nonnegative curvature operator, Math. Ann., 337(2) (2007), 435-442.

X. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc, 136(11) (2008), 4075-4078.

X. Cao, J. Guckenheimer and L. Saloff-Coste, The backward behavior of the Ricci and cross curvature flows on SL (2; R), Comm. Anal. Geom., 17(4) (2009), 777-796.

X. Cao and L. Saloff-Coste, Backward Ricci flow on locally homogeneous three-manifolds, Comm. Anal. Geom., 12(2), (2009) 305-325.

X. Cao, S. Hou and J. Ling, Estimates and monotonicity of the first eigenvalue under the Ricci flow, Math. Ann. 345 (2012), no. 2, 451-463.

D. Friedan, Nonlinear models in 2+ dimensions, Annals of physics 163 (31) (1985), 318-419.

R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255-306.

S. Hou, Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds, Acta Mathematica Sinica, English series, 136(11) (2018), 1179-1194.

S. Hou, Eigenvalues under the Ricci flow of model geometries, (Chinese) Acta Math. Sinica (Chin. ser.) 60 (2017), no. 4, 583-594.

F. Korouki and A. Razavi, Bounds for the first eigenvalue of (−∆ − R) under the Ricci flow on Bianchi classes, Bull. Braz. Math. Soc, (2019).

J. Li, Eigenvalues and energy functionals with monotonicity formula under Ricci flow, Math. Ann., 338 (2007), 927-946.

J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math., 21(3) (1976), 293-329.

G. Perelman, The entropy formula for the Ricci flow and it’s geometric applications, Arxiv (2002).

L. Wang, Eigenvalue estimate for the weighted p-Laplace, Annali di Matematica, 191 (2012), 539-550.

L. Wang, Gradient estimates on the weighted p-Laplace heat equation, J. Diff. Equ., 264 (2018), 506-524.

J. Wu, E. Wang and Y. Zheng, First eigenvalue of the p-Laplace operator along the Ricci flow, Ann. Glob. Anal. Geom., 38(1) (2009), 27-55.

J. Wu, First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow, Acta Mathematica Sinica, English series, (2011), 1591-1598




DOI: https://doi.org/10.22342/jims.26.3.934.380-392

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