Eigenvalues variation of the p-Laplacian under the Yamabe flow

Abstract: The problem of geometric flow and determining the eigenvalues for nonlinear operators acting on finite-dimensional manifolds is a known problem. In this paper we will consider the eigenvalue problem for the p-Laplace operator acting on the space of functions on closed manifolds. We find the first variation formula for the eigenvalues of p-Laplacian on a closed manifold evolving by the Yamabe flow and find some applications.


Introduction
Let (M, g) be a compact Riemannian manifold. Studying the eigenvalues of geometric operators for instance p-Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings. This problem has a wide range of applications and is one of the main tools for dealing with such linear and nonlinear operators. There are many mathematicians who investigate properties of the spectrum of Laplacian and estimate the spectrum in terms of the other geometric quantities of M (see Cheng, 1975;Cheng & Yang, 2005;Harrell II & Michel, 1994;Leung, 1991).

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Let (M, g) be a compact Riemannian manifold. Studying the eigenvalues of p-Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings.
In this paper we will consider the eigenvalue problem for the p-Laplace operator acting on the space of functions on closed manifolds. We find the first variation formula for the eigenvalues of p-Laplacian on a closed manifold evolving by the Yamabe flow and find some applications.
In Yamabe (1960) proposed a generalization of the uniformization theorem to higher dimensions. In Hamilton (1982) proposed a heat flow approach to the Yamabe problem. To describe this, let g(t) be a smooth one-parameter family of Riemannian metrics on M. Metric g(t) is a solution of the unnormalized Yamabe flow if where R g(t) denotes the scalar curvature of g(t). It is often convenient to consider a normalized version of the flow. Also, g(t) is a solution of the normalized Yamabe flow if Here, r g(t) denotes the mean value of the scalar curvature of the metric g(t); that is, The evolution Equations (1.1) and (1.2) are equivalent in the sense that any solution of the Equation (1.1) can be transformed into a solution of (1.2) by a rescaling procedure.
There were elaborated a set of applications with such geometric flows, following low dimensional or approximative methods to construct solutions of evolution equations, in modern gravity and mathematical physics, for instance, for gravity (Nitta, 2005), black holes (Headrick & Wiseman, 2006), and mechanics and classical field theory (Miron & Anastasiei, 1994, 1997.

Eigenvalues of p-Laplacian
Let M be a closed Riemannian manifold and f be a smooth scalar function on M. The p-Laplacian of f for 1 < p < ∞ is defined as where and in local coordinate, we have Let (M n , g) be a closed Riemannian manifold. We say that is an eigenvalue of the p-Laplace operator whenever for some f ∈ W 1,p (M), or equivalently Normalized eigenfunctions are defined as follows : Vol(M) . (1.3) In what follows we assume, the existence and C 1 -differentiability of the elements (t) and f(t) under a Yamabe flow deformation g(t) of a given initial metric. We prove some facts about the spectrum variation under a deformation of the metric given by a Yamabe flow equation. A similar geometric problem have been considered from a different point of view (see Cao, 2007Cao, , 2008Cerbo, 2007;Perelman, 2002;Wu, 2011).

Variation of (t)
In this section, we will give some useful evolution formulas for (t) under the Yamabe flow.
) be a solution of the un-normalized Yamabe flow on the smooth closed manifold (M n , g 0 ). If (t) denotes the evolution of an eigenvalue under the Yamabe flow, then where f is the associated normalized evolving eigenfunction. (2.5) Now, using (1.4), from the condition (2.7) implies that Replace (2.8) in (2.6), we have: square Now, we give a variation of (t) under the normalized Yamabe flow which is similar to the previous proposition.
Proposition 2.2 Let (M n , g(t)) be a solution of the normalized Yamabe flow on the smooth closed manifold (M n , g 0 ). If (t) denotes the evolution of an eigenvalue under the Yamabe flow, then where f is the associated normalized evolving eigenfunction.
Proof In the normalized case, the integrability conditions read as follows: Since we can then write Hence the proposition is obtained by replacing (2.13) in (2.12). ✷

Variation of (t) on a surface
Now, we write Proposition (2.1) and (2.2) in some remarkable particular cases. (2.12) (2.14) ) be a solution of the normalized Yamabe flow on a compact surface then from (Chow & Knopf, 2004)  Proof From Chow and Knopf (2004), under the un-normalized Yamabe flow on a surface, we have by the maximum principle, the nonnegativity of the scalar curvature is preserved along the Yamabe flow. Then (2.14) implies that d (

Variation of (t) on homogeneous manifolds
In this section, we consider the behavior of the spectrum when we evolve an initial homogeneous metric.
Let (M n , g(t)) be a solution of the un-normalized Yamabe flow on the smooth closed homogeneous manifold (M n , g 0 ). If (t) denote the evaluation of an eigenvalue under the Yamabe flow, then the evolving metric remains homogeneous. On the other hand a homogeneous manifold has constant scalar curvature. Therefore (2.1) implies that If we suppose that (M n , g(t)) is a solution of the normalized Yamabe flow on the smooth homogeneous closed manifold (M n , g 0 ), then (2.9) implies that (t) = (0).

Variation of (t) on 3-dimensional manifolds
In this section, we consider the behavior of (t) on 3-dimensional manifolds.
Proposition 2.8 Let (M n , g(t)) be a solution of the normalized Yamabe flow on a closed manifold whose Ricci curvature is initially negative and there exists ≥ 0 such that then Proof From Surez-Serrato and Tapie (2012), for any solution of the Yamabe flow on a closed manifold with negetive curvature, we have −c < R(., t) < − < 0. Hence the proposition is established by using (2.9), which implies that ✷

Examples
In this section, we show that the variational formula is effective to derive some properties of the evolving spectrum of p-Laplace operator and then we find (t) for some of Riemannian manifolds.
Example 3.1 Let (M n , g 0 ) be an Einstein manifold i.e. there exists a constant a such that Ric(g 0 ) = ag 0 . Assume that we have a solution to the Yamabe flow which is of the form where u(t) is a positive function. We compute for this to be a solution of Yamabe flow, we require: this shows that: therefore: (2.16) g(t) = u(t)g 0 , u(0) = 1 g t = u � (t)g 0 u � (t)g 0 = −Rg = −g ij Ric(g ij )g = − 1 u(t) (g 0 ) ij Ric((g 0 ) ij )u(t)g 0 = −R 0 g 0 = −ang 0 u � (t) = −na,