Eigenvalues of the p-Laplacian and Evolution under the Ricci-Harmonic Map Flowc

Properties of eigenvalues of the p-Laplacian operator on a finite dimensional compact Riemannian manifold are studied for the case in which the metric of the manifold evolves under the Ricci-harmonic map flow. It will be shown that the first nonzero eigenvalue is monotonically nondecreasing along the flow and differentiable almost everywhere.


x t x t x t x t t x t x t t
,  denotes tensor product, Ric is the Ricci cur- vature tensor corresponding to g and   0 The problem to be investigated here is the p -eigenvalue problem where   .  The objective is to present a new concise proof of the general evolution of the first eigenvalue as a function of t under the Ricci flow (1.1).The proof is based on the work of Cao [6] [7] and Abolarinwa [8].A monotonicity formula without differentiability assumption on the eigenfunction can also be obtained.The differentiability of a p -eigenvalue is a consequence of the monotonicity for-mula.
For the most part, a local coordinate system   The fact that the Riemannian metric is parallel, 0 g , will be used frequently without further mention as well as integration by parts, which for example takes the form, Also the following notations for the Ricci-harmonic map flow [1] will be used in the following form,

The Ricci Flow
All the geometric quantities associated with the manifold M evolve as the Riemannian metric on M evolves along the Ricci-harmonic map flow.
Lemma 1.Let a one-parameter family of smooth metrics   gt solve the Ricci-harmonic map flow (1.1).Then the following evolutions hold: Here w is a smooth function defined on M and g S the metric trace of the symmetric 2-tensor ij S as in (1.6).Proof: To prove equation (2.1), recall the metric satisfies ij i jl l gg   .Differentiating both sides of this with respect to t and using (1.1), we have To obtain the second result (2.2), differentiate To obtain (2.3), differentiate both sides of the volume form on M with respect to t to obtain, By the chain rule, we get,   To obtain the results for the p -Laplacian, the following Lemma will be very important.
Lemma 2. Suppose a one-parameter family of smooth metrics   gt solves Ricci-harmonic map flow (1.1).Then there are the following evolutions

Replace
p by 2 p  in (a) and the result in (b) follows immediately.(c)

Study of the Eigenvalue Problem
A nonlinear eigenvalue problem is introduced which involves the p -Laplacian (1.3) and is defined as with 0 u  and subject to the normalization condition d 1.
One of the main objectives is to derive a general evolution equation for the p -eigenvalues of the p -Laplacian.Out of this, it can be shown that ,1 p  is monotone on those metrics which evolve under the Ricci-harmonic map flow.
The continuity and differentiability of ,1 p  can be derived from its evolution by using Cao's approach.To study this, begin by multiplying (3.1) by the function u on both sides and then integrating over M using (3.2) to obtain The function u will satisfy the following integrability condition d 0.
This can be developed by direct computation, By the third part of Lemma 2, by the evolution of , pg  , the first part of this takes the form, Integrating the second and third terms in I by parts gives Computing the first term in I , we get Therefore, putting all of these into (3.9)for the time derivative, it has been found that Therefore, the result simplifies considerably to the form, The last pair of integrals can be simplified in the following way, The first two terms cancel out and so integrating the last term by parts using the definition    , which is the usual Laplace-Beltrami operator.Thus this theorem implies that the first eigenvalue of  and the corresponding eigenfunction are smoothly differentiable for this operator as well.

Evolution of the First Eigenvalue
There are some important consequences of Theorem 1 with regard to the evolution of the first eigenvalue that will be discussed now.
Corollary 1.Under the conditions of Theorem 1, it is the case that where  is defined to as This has the implication that the function  

,
Nh be two compact Riemannian manifolds without boun-dary with dimensions m and n , respectively.Let : MN   be a smooth map that is a critical point of the Dirichlet energy integral is the integration measure on the manifold.Nash's embedding theorem implies N is isometrically embedded in d R for dn  .The configuration and a family of smooth maps   , xt  is defined to be a Ricci-harmonic map flow if it satisfies the coupled system of nonlinear parabolic equations

1 w
is the energy minimizer of the p -Rayleigh quotient (1.4) such that the infimum runs over all The Riemannian structure on the manifold M allows a Riemannian volume measure

3 )
Integrating this by parts once, it follows that 4) implies that the eigenvalues from (3.1) are all positive.Suppose now that   , u x t is the eigenfunction that corresponds to the first

6 )
At this point, it is possible to prove a theorem regard to the evolution, monotonicity and differentiability of the first eigenvalue of the p -Laplacian under the Ricci-harmonic map flow.Theorem 1.Let   , Mg be an m -dimensional, closed Riemannian manifold evolving by the Ricci-harmonic map flow.Let

Theorem 2 .
Ricci-harmonic map flow, and this is important enough to be summarized in the form of Theorem 2. Under the assumptions of Theorem 1, the function .
Proof: Integrating both sides of the inequality Completing the integral on the left, this immediately gives (4.3).□  