1. Introduction
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 [1] showed that a functional depending on scalar curvature is nondecreasing along the Ricci flow coupled to a type of heat equation [2] [3]. 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 [4] [5]. There is a generalization of the p-Laplacian to a class of
-Laplacian which has applications in applied mathematics and physics [6] [7].
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
(1)
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,
(2)
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
(3)
The variable
in (3) is called the average of
with respect to
, and defined as
(4)
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,
(5)
where we have set
. The p-Laplacian of f is defined for
as
(6)
where
, where
are vector fields on
. In local coordinates,
(7)
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
(8)
In (8),
and
and
are positive real numbers which satisfy the condition
(9)
It is said that
in (8) is an eigenvalue for the system whenever for some
and
it is the case that
(10)
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
(11)
In (11),
and
are defined to be
(12)
Let
be a solution of the
flow (1) on the smooth manifold
with
. Then
(13)
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
(14)
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
(15)
The integrand is given by
(16)
Proof: Let us put
(17)
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,
(18)
Furthermore, functions
can be defined along this flow according to the equations
(19)
In (19), the functions
and
are smooth functions under the Ricci harmonic flow and they satisfy the condition
(20)
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
(21)
With (21), (3) can be expressed in components using (14) as
(22)
Hence, if f is a smooth function with respect to t, then along the Ricci-harmonic flow, we find that
(23)
Substituting the result from (22), we have
(24)
The measure
also depends on t through g and has derivative
(25)
Since
and
are smooth functions so too is
with respect to t. Let us write
(26)
Using (24) and (25) with f replaced by u and v, it follows that,
(27)
Integrating both sides of (26) with respect to t between
and
, it follows that
(28)
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
(29)
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
(30)
Thus (16) is given by
(31)
Differentiating the normalization condition and using (25), we get
(32)
The results in (10) imply that by replacing function
by
and
by
, one obtains
(33)
Multiply the first equation in (33) by
and the second by
and then add the two, then we obtain that
(34)
Multiply (32) by
and then subtract the resulting expression from (34),
(35)
Substituting (35) into
given in (31), we have
(36)
Substitute the hypothesis given in (29) into (36) to yield the inequality
(37)
Using the definition of
from (14) and the two known results
(38)
it follows that since
the last term in (38) is positive the lower bound results
(39)
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
(40)
is exactly the function
(41)
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,
(42)
In any small neighborhood of
then it also holds that
. So it follows that for any
sufficiently close to
,
(43)
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
(44)
In terms of u and v, let us introduce the function
(45)
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
(46)
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
(47)
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
(48)
To get the right-hand side, Equation (25) has to be modified to
(49)
Hence the t derivative of
is given by
(50)
For the normalized Ricci flow, the following relation holds,
(51)
Now replacing (51) in (50), we obtain the result
(52)
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
(53)
on
with
. Then the quantity
is nondecreasing along the flow on
where
.
Proof: It has been shown that
(54)
Using condition (53), the following bound is produced
(55)
If
then (41) implies that positivity of S persists under this type of flow. Using (20) we have therefore,
(56)
Then in any sufficiently small neighborhood about the value
, it can be concluded that
(57)
This is a separable equation, so integrating inequality (57) with respect to t over the interval
,
(58)
Integrating we obtain
(59)
Since
and
, it is concluded that
(60)
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,
(61)
Consequently, we can calculate that
(62)
For any vector w, then we can contract
with the
to get
(63)
If
where
, then
, hence using
and simplifying, we have
(64)
Now with
and
,
(65)
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
(66)
The solution for
of the initial value problem is given by
(67)
The solution of the flow remains Einstein and so we have,
(68)
By (46), we find that
(69)
If it is assumed that
then for
and
where
, we have
(70)
In any sufficiently small neighborhood of
,
(71)
Integrating this inequality with respect to t on
, we find that
(72)
As
is arbitrary,
and
. It can be concluded from this that
is nondecreasing along the Ricci-harmonic flow on
.
5. Summary
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.