FIRST EIGENVALUE OF p -LAPLACIAN ALONG THE NORMALIZED RICCI FLOW ON BIANCHI CLASSES

. 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.


Introduction
Over the last few years, studying the geometric flows, specially the Ricci flow have become a topic of active research in both mathematics and physics. Generally, a geometric flow is an evolution of a geometric structure under a differential equation related to a functional usually associated with a curvature in a manifold. Although, the Ricci flow was introduced first by Hamilton [10] in mathematics and in the work of Friedan [9] in the context of string theory, Perelman has made a current wide interests by the proof of Poincare' conjecture using the Ricci flow in [16]. Consider M as a manifold with Riemannian metric g 0 , the family g(t) of Riemannian metrics on M have been called as an un-normalized Ricci flow when it satisfies the equation where Ric is known Ricci tensor of g(t).
And also one can consider the normalized Ricci flow as follow d dt g(t) = −2Ric (g(t)) + 2r n g g(0) = g 0 , where r = M Rdµ M dµ is the average of scalar curvature. Hamilton in [10], has shown that there is a unique solution for the Ricci flow (1), on the interval [0, T ) for a sufficient T . Now consider g(t) as a solution of the Ricci flow (1), the customary normalization on 3-manifolds is settinḡ n where n = 3 and r is as same as what mentioned above is average of scalar curvature. In this caseḡ(t) will be the solution of normalized Ricci flow (2). In [16], Perelman has shown that the first eigenvalue of −∆ + R 4 is nondecreasing under the Ricci flow. Later Cao [4] has shown the similar result for the eigenvalues of −∆ + R 2 on a manifolds with non-negative curvature operator. Also similar results hold for the first eigenvalue of −∆ + aR a ≥ 1 4 along the Ricci flow, for more details see [5,14]. There are some other published work in monotonicity of eigenvalues of geometric operators under some geometric flows. For example second author in [2], has studied the evolution for the first eigenvalue of p-Laplacian along the Yamabe flow and also in [3] shown the monotonicity of eigenvalues of Witten-Laplace operator along the Ricci-Bourguignon flow. Also for more details in a case of p-Laplacian operator, Wang in [17], has shown the eigenvalue estimate for the weighted p-Laplacian and later in [18] shown the gradient estimate on the weighted p-Laplace heat equation. Beside what mentioned before A. Abolarinwa in [1], has studied the evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow and also you can find some useful results in eigenvalue monotonicity of the p-Laplace operator under the Ricci flow in [20], also Cao and Songbo Hou have worked on monotonicity of the first eigenvalue under Ricci flow and you can see their results in [8,12]. Finally we will use some results of [19] in this work. In this paper we will investigate the evolution of the first eigenvalue of p-Laplacian operator and then we will find some bounds in a case of Bianchi classes.

Preliminaries and evolution equation
Let (M, g) be a locally homogeneous closed 3-manifold, there are nine classes of such manifolds. They are divided into two groups, the first consists of H(3), H(2)×R 1 and So(3)×R 1 and the other one includes R 3 , SU (2), SL(2, R), Heisenberg, E(1, 1) and E(2) which are called Bianchi classes. Milnor [15], has provided a frame where both the metric and Ricci tensors are diagonalized and this property is preserved by the Ricci flow (1). Now let {θ} 3 i=1 be a dual to Milnor's frame, we consider the metric g(t) as then the Ricci flow becomes a system of ODE with three variables {A(t), B(t), C(t)}. Consider M as a compact Riemannian manifold and u : M −→ R be a smooth function on M or we can consider u ∈ W 1,p (M ) the Sobolev space. The p-Laplacian of u for 1 < p < ∞ is defined as In this case we say that λ is an eigenvalue of p-Laplace operator whenever there exist a function u on M such that The theorem below from [19] gives us the continuity of the first eigenvalue of p-Laplace operator.
Theorem 2.1. If g 1 and g 2 are two metrics which satisfy then for any p > 1, we have In particular λ(g(t)) is a continuous function in a t-variable.
Discussing about the monotonicity of the first eigenvalue of the p-Laplace operator powerfully is dependent to the differentiability of the eigenvalue function. In this section we are following the X. Cao's argument [4], where we introduce the smooth eigenvalue function λ(u, t) which is smooth then we can write the monotonicity of λ(t). We assume at time t 0 , u 0 = u(t 0 ) is eigenfunction for the first eigenvalue λ(t 0 ) of p-Laplacian. Then we have M |u(t 0 )| p dµ g(t0) = 1.
We consider the following smooth function and normalize this smooth function as under the Ricci flow. Now we define a general smooth function as where u is any smooth function satisfying In general λ(u, t) is not equal to λ(t). But at time t 0 we conclude that Now we are ready to give an evolution formula for λ(u, t)| t=t0 along the normalized Ricci flow in a case of 3-homogeneous manifold. In this case it is not hard to see R = r where R is scalar curvature and r is as same as what explained in the definition of the normalized Ricci flow (2).
) be a solution of the normalized Ricci flow (2), on a locally homogeneous 3-manifold. If λ 1,p (t) denotes the first eigenvalue of the p-Laplacian (3), then Proof. By the direct computation it will be easy to see that under the normalized Ricci flow (2) we have and also ∂ ∂t The function λ(u, t) is smooth so it concludes that As we mentioned before in homogeneous manifold R = r also the condition M |u| p dµ = 1 results that which implies what we looking for.

Estimate of λ(t) on Bianchi classes
In this section we are going to give some useful bounds for λ(t) separately in Bianchi classes. Before Hou in [11] has given bounds for the first eigenvalue of ∆ in a case of u > 0 under the backward Ricci flow and also he proved the eigenvalue evolves toward zero in a case that the backward Ricci flow converges to a sub-Riemannian geometry by a proper rescaling. Later Razavi and Korouki in [13] have done similar work for the first eigenvalue of (−∆ − R) under the Ricci flow.

Remark 3.1. In homogeneous condition and in a case of the un-normalized Ricci flow we get that
Now we study the behavior of the first eigenvalue of p-Laplacian in each classes separately. Case 1: R 3 In this case all metrics are flat, so for all t ≥ 0 we have g(t) = g 0 where g 0 is initial metric, therefore λ(t) is constant.

Case 2: Heisenberg
This class is isomorphic to the set of upper-triangular 3 × 3 matrices endowed with the usual matrix multiplication. Under the metric g 0 we choose a frame also under the normalization A 0 B 0 C 0 = 1 we have Proof. Under the proposition 2.2 and Ricci coordinates in Hiesenberg case, we get By substituting R into formula (7) we obtain Since λ(f, t) is smooth function with respect to time t, hence in any sufficiently small neighborhood of t 0 , we have Since t 0 is arbitrary then for any t ∈ [0, T ) the inequality (8) holds and it implies Therefore the quantity λ(t)e t 0 (− 2 3 pA 2 )dτ is nondecreasing. Also in a similar way we have Which implies what we are looking for.

Remark 3.3.
In this case by [11], for the tensors A,B and C we have , where R 0 = − 1 2 A 2 0 . Now by substituting these formulas into the formula (7) which is hold for arbitrary t 0 , and integrating from both sides in [t 0 , t], we get and similarly

Case 3: E(2)
Manifold E(2) is the group of isometries of Euclidian plane. In this case we have an Einstein metric and Ricci flow converges exponentially to flat metrics. Dependent to the metric g 0 we choose the frame In this case under the normalization A 0 B 0 C 0 = 1 we have X. Cao in [7] has proved that for initial tensors A 0 and B 0 Proof. Since by [6] we have A > B and also under the proposition 2.2, we get where by the assumption A 0 > B 0 we have In a similar way Rλ(t 0 ), now apply R from above, since t 0 is arbitrary it implies what mentioned before in the theorem.
Also under the normalization A 0 B 0 C 0 = 1 we obtain dt and λ(t)e t 0 ( 2 3 p(A+C) 2 )dt are non-decreasing and non-increasing respectively, where Proof. In the case of A 0 = C 0 by [6] we get that A = C, it is easy to see Proof. By X. Cao [7], the proof of first and second section will be easy and similarly for the third section we have Also in a similar way which if you substitute R, it is completed the proof.
Case 6: SL(2,R) On SL (2, R) there is no Einstein metric and the Ricci flow doesn't converge and develops a pancake degeneracy, also by the frame {X i } 3 i=0 , we get [X 2 , X 3 ] = −X 1 , [X 3 , X 1 ] = X 2 , [X 1 , X 2 ] = X 3 , in this case we also have In which