The eigenvalue problem which arises from geometric operators under various kinds of geometric flows has attracted a great deal of attention recently, as it is a very effective method of studying Riemannian manifolds. This area of investigation opened up when Perlman  showed that a functional depending on scalar curvature is nondecreasing along the Ricci flow coupled to a type of heat equation  . This property of the functional implies that the first eigenvalue of a geometric operator is nondecreasing under Ricci flow. The geometric operator has also been studied with regard to its eigenvalues along the Ricci flow and Ricci-Bourguignon flow as well. The evolution of the first eigenvalue has been looked at in the case of the p-Laplacian along a Ricci-harmonic flow, and the Ricci flow and the m-th mean curvature flow respectively  . There is a generalization of the p-Laplacian to a class of -Laplacian which has applications in applied mathematics and physics  .
A geometric flow is an evolution of a geometric structure relevant to a given manifold. Let be a closed, m-dimensional Riemannian manifold that has metric . Hamilton first introduced the Ricci flow by means of the differential equation
In (1), is an evolution parameter and is the Ricci tensor of metric . Lowered indices are understood to apply in (1) so that . Let and be two closed Riemannian manifolds. By Nash’s embedding theorem, it may be assumed that is isometrically embedded into Euclidean space . Identify maps with for sufficiently large . Then a generalization of Ricci flow can be established as follows,
In (2), is a positive constant, is a family of smooth maps from to a closed target manifold and is the intrinsic Laplacian of which denotes the tension field of with respect to the evolving metric . This system of evolution equations will be called the Ricci flow coupled to a harmonic flow. It has been shown that (2) has a unique solution with the initial data . It is also useful to define a normalized Ricci-harmonic flow defined as
The variable in (3) is called the average of with respect to , and defined as
When integrating over the manifold, we simply write , and is the volume form or measure on . Under normalized Ricci flow, the volume of the solution metrics remains constant with respect to t.
2. Definition of the Eigenvalue Problem
Let be a closed Riemannian manifold and let be a smooth function on the manifold so suppose . The Laplace-Beltrami operator which acts on a smooth function f defined on is the divergence of the gradient of f,
where we have set . The p-Laplacian of f is defined for as
where , where are vector fields on . In local coordinates,
When the p-Laplacian becomes the Laplace-Beltrami operator. Let be a closed Riemannian manifold. To present the problem, consider the following nonlinear system of equations on
In (8), and and are positive real numbers which satisfy the condition
It is said that in (8) is an eigenvalue for the system whenever for some and it is the case that
The functions , and is the closure of in the Sobolev space . The set of functions are called the eigenfunctions which correspond to the eigenvalue . A first positive eigenvalue of (8) can be determined by computing
In (11), and are defined to be
Let be a solution of the flow (1) on the smooth manifold with . Then
defines the solution of an eigenvalue of (8) under the variation of . The eigenfunctions associated to are normalized such that .
The first eigenvalue of a class of -Laplacians given in (8) is studied such that its metric satisfies the flow. Let us denote differentiation with respect to t as , and introduce tensor and its trace
where R is the Ricci scalar curvature.
3. Variational Formulation
Some useful evolution equations for under the Ricci harmonic flow will be formulated. In particular, a useful result concerning the variation of the first eigenvalue (8) under the Ricci harmonic flow is considered next.
Theorem 1: Let with be a solution of the Ricci harmonic flow on the closed manifold . Let be the first eigenvalue of the -Laplacian along this flow. For any such that , we have
The integrand is given by
Proof: Let us put
suppose that at we assign be the eigenfunctions corresponding to the eigenvalue for the -Laplacian. Define the following smooth functions along the Ricci harmonic flow as follows,
Furthermore, functions can be defined along this flow according to the equations
In (19), the functions and are smooth functions under the Ricci harmonic flow and they satisfy the condition
Now are the eigenfunctions of the eigenvalue for the -Laplacian at time , that is, . The following formula will be needed, which arises from the fact that
With (21), (3) can be expressed in components using (14) as
Hence, if f is a smooth function with respect to t, then along the Ricci-harmonic flow, we find that
Substituting the result from (22), we have
The measure also depends on t through g and has derivative
Since and are smooth functions so too is with respect to t. Let us write
Using (24) and (25) with f replaced by u and v, it follows that,
Integrating both sides of (26) with respect to t between and , it follows that
where and . since it is the case that , then setting in (28), it is seen that (15) follows immediately with satisfying (16).
Theorem 2: Let be a solution of the Ricci-harmonic flow on the smooth, closed manifold and let denote the evolution of the first eigenvalue under the flow. Suppose that and on it holds that
If then is nondecreasing and differentiable almost everywhere along the Ricci-harmonic flow (2) on
Proof: For any let be the eigenfunctions corresponding to the value of the -Laplacian. Then there is the normalization condition
Thus (16) is given by
Differentiating the normalization condition and using (25), we get
The results in (10) imply that by replacing function by and by , one obtains
Multiply the first equation in (33) by and the second by and then add the two, then we obtain that
Multiply (32) by and then subtract the resulting expression from (34),
Substituting (35) into given in (31), we have
Substitute the hypothesis given in (29) into (36) to yield the inequality
Using the definition of from (14) and the two known results
it follows that since the last term in (38) is positive the lower bound results
Thus S is a supersolution of the partial differential equation . To be able to use the maximum principle, it has to be observed that the solution to the equation
is exactly the function
for , where . Applying the maximum principle to (39), it must be that along the Ricci-harmonic flow. If the nonnegativity of S is preserved along the flow and (37) has the property,
In any small neighborhood of then it also holds that . So it follows that for any sufficiently close to ,
Since is arbitrary, the first part of the claim is complete. For differentiability of note that as is increasing and continuous on the interval , the Lebesgue theorem implies that function is differentiable almost everywhere on . Thus the proof is complete.
4. Ricci Flows
A smooth eigenvalue function can be introduced along the Ricci harmonic flow. Evolution equations can be developed for this. Let be an m-dimensional closed Riemannian manifold and let be a smooth solution of the flow. Introduce a function which depends on u, v and which satisfy the three integral constraints
In terms of u and v, let us introduce the function
With respect to the variable t, is a snooth eigenvalue type function. In the case where are the corresponding eigenfunctions corresponding to the first eigenvalue , then . In this case, (45) gives the eigenvalue directly without going through the process indicated in (11). This leads us to formulate the following Proposition which can be proved along exactly the same lines as the two proceeding results.
Proposition 1: Let be a solution of the Ricci harmonic flow on the smooth closed manifold . If denotes the evolution of the first eigenvalue under this flow, then
Here u and v are the associated normalized evolving eigenfunctions.
At this point we can start to study the evolution of under the normalized flow (3), which is similar to what has already been done.
Theorem 3: Let be a solution of the normalized Ricci harmonic flow on a smooth closed manifold . If denotes the evolution of the first eigenvalue under the flow (3), then
in which are the associated evolving, normalized eigenfunctions for the problem.
Proof: In the normalized case start by differentiating the first integrability condition in (44) with respect to the parameter to find
To get the right-hand side, Equation (25) has to be modified to
Hence the t derivative of is given by
For the normalized Ricci flow, the following relation holds,
Now replacing (51) in (50), we obtain the result
The first term of the third line in result (52) is just , so this term cancels with the second in that same line and what remains is exactly the desired result (47).
Theorem 4: Let be a solution of the Ricci harmonic flow on the smooth closed manifold and denotes the evolution of the first eigenvalue under the flow. If and
on with . Then the quantity is nondecreasing along the flow on where .
Proof: It has been shown that
Using condition (53), the following bound is produced
If then (41) implies that positivity of S persists under this type of flow. Using (20) we have therefore,
Then in any sufficiently small neighborhood about the value , it can be concluded that
This is a separable equation, so integrating inequality (57) with respect to t over the interval ,
Integrating we obtain
Since and , it is concluded that
This implies that the function is nondereasing on any sufficiently small neighborhood of . Since is arbitrary, we conclude that is nondecreasing along this flow over the interval .
Now if is taken to be zero, then the Ricci harmonic flow reduces to the Dirac flow and the theorem implies that the following Corollary can be stated.
Corollary 1: Let for be a closed Riemannian manifold such that denotes the first eigenvalue of the -Laplacian With and condition (53) in effect along the Ricci flow, then (a) If then is nondecreasing along the Ricci flow for any . (b) If then the quantity is nondecreasing along the Ricci flow for any where we define .
If we simply work with two-dimensional manifolds or surfaces, then the following result must hold.
Theorem 5: Let with be a solution of the Ricci harmonic flow on a closed Riemannian surface and let denote the first eigenvalue of the -Laplacian (8). (1) Suppose that and along the Ricci flow. If , the function is nondecreasing along the flow for any . If , the function is nondecreasing along the Ricci-harmonic flow on where . (2) Suppose that . If then is nondecreasing along the Ricci-harmonic flow for any . If the quantity is nondecreasing along this flow on where
Proof: In the case of two dimensional manifolds, the tensor Ric takes the simple form,
Consequently, we can calculate that
For any vector w, then we can contract with the to get
If where , then , hence using and simplifying, we have
Now with and ,
To get the second last inequality in (65), use has been made of in (61) for the two-dimensional case. The result now follows by using Theorems 2 and 4.
Corollary 2: Let , be a solution of the Ricci flow on a closed Riemannina surface and denotes the first eigenvalue of the -Laplacian (8). (1) If then is nondecreasing along the Ricci flow for any . (2) If , then the quantity is nondecreasing along the Ricci flow for any where .
As an illustration of these ideas, let be an Einstein manifold so there exists a constant such that and suppose so is the identity map. Assuming that , is a function and the fact is a harmonic map for all , then the Ricci-harmonic flow reduces to
The solution for of the initial value problem is given by
The solution of the flow remains Einstein and so we have,
By (46), we find that
If it is assumed that then for and where , we have
In any sufficiently small neighborhood of ,
Integrating this inequality with respect to t on , we find that
As is arbitrary, and . It can be concluded from this that is nondecreasing along the Ricci-harmonic flow on .
The main results here have been to define a p-Laplacian eigenvalue problem and to find a way to study the evolution of the first eigenvalue under the Ricci flows established in Equations (2) and (3). It has also been found that flows for some related quantities can also be studied. This work will provide a foundation for the study of similar problems in the future.