Riemannian manifolds with positive Yamabe invariant and Paneitz operator
For a Riemannian manifold with dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.
Let be a smooth compact Riemannian manifold with dimension . Denote as the conformal class of metrics associated with . The Yamabe invariant is given by (see [LP])
here is the scalar curvature of and is the measure associated with . In terms of the conformal Laplacian operator
In particular if and only if the first eigenvalue . Moreover based on the fact that the first eigenfunction of must be strictly positive or negative, we know
For a Riemannian manifold with dimension at least five, can we find a conformal invariant condition which is equivalent to the existence of a conformal metric with positive scalar and curvature?
In view of the results on positivity of Paneitz operator in [GM, XY], many people suspect the conformal invariant condition wanted in Problem 1.1 should be the positivity of Yamabe invariant and Paneitz operator. As a consequence of the main result below, this is verified for dimension at least six (see Corollary 1.1). It is very likely the statement is still true for dimension five. However due to technical constrains in our approach, dimension five case remains an open problem.
To write down the formula of curvature and Paneitz operator, following [B], let
here is the Ricci curvature tensor. The curvature is given by
The Paneitz operator is given by
Here is a local orthonormal frame with respect to . For , under a conformal change of metric, the Paneitz operator satisfies
We now define two conformal invariants related to the -curvature. First, in analogy with the Yamabe invariant, we define
We use to emphasize the infimum is taken over positive functions (i.e. conformal factors). To define the second invariant, denote
It is clear that is well defined for . Define
but in general (and in contrast to the usual Yamabe invariant) we have an inequality instead of equality, due to the fact Paneitz operator is fourth order. Note that from standard elliptic theory if and only if the first eigenvalue i.e. is positive definite.
When the Yamabe invariant , there is another closely related quantity defined by
One of the main goals of this paper is to understand the relationship between these quantities, and their connection to the existence of a metric with positive scalar and -curvature.
Our motivation for studying these problems comes from recent progress in the understanding of curvature equations for dimension at least five in [GM, HY1, HY2]. Paneitz operator and curvature were brought to attention in [CGY]. For dimension at least five, in various work [HeR, HuR, R], people had realized the important role of positivity of Green’s function of Paneitz operator in understanding the curvature equations. Such kind of positivity is hard to get due to the lack of maximum principle for fourth order equations. A breakthrough was achieved in [GM], namely for , if and , then and the Green’s function of Paneitz operator is strictly positive. Subsequently in [HY1], for positive Yamabe invariant case it was found the positivity of Green’s function of Paneitz operator is equivalent to the existence of a conformal metric with positive curvature. Indeed it was shown that if , , then there exists a conformal metric with positive curvature if and only if and , it is also equivalent to and for a fixed . Note the positivity of Green’s function is a conformal invariant condition. In [GM], it was shown that and implies is achieved at a positive smooth function. The assumption was relaxed to and in [HY2]. Trying to understand relations between various assumptions motivates problems considered here.
Let be a smooth compact Riemannian manifold with dimension . If and , then there exists a metric satisfying and .
Let be a smooth compact Riemannian manifold with dimension . Then the following statements are equivalent
there exists a metric satisfying and .
Let be a smooth compact Riemannian manifold with dimension . If and , then , the Green’s function , and is achieved at a positive smooth function with and . In particular,
The dimensional restriction is an unfortunate by-product of our technique and it is very likely the result holds in dimension five as well. To explain our approach, we first point out that the curvature equation is variational: a metric has constant curvature if and only if it is a critical point of the total curvature functional with running through the set of conformal metrics with unit volume. A closely related quantity is
Indeed equation is also variational in similar sense. Since
Obviously is always nonnegative. For consider the functional
A critical metric of this functional restricted to the space of conformal metrics of unit volume satisfies
In the appendix, we will use elementary methods to show that if , then there exists , and such that
and . Define as
Then for we consider the following 1-parameter family of equations:
Then . We will show is both open and closed by the implicit function theorem and apriori estimates. Hence . It follows that there exists a with and .
We conclude the introduction with some remarks. The dimensional restriction appears in both the open and closed part of the argument. The power of the conformal factor on the right hand side of (1.22) is chosen to be negative to give better estimate of solutions. Moreover this choice also leads to a good sign of the zeroth order term in the linearized operator. We also observe that our path of equations is variational, it has a divergence structure which we will exploit in apriori estimates. At last we note that in dimension four a path of equations which is analogous to (1.22) is considered in [CGY] to produce a conformal metric with positive scalar curvature and curvature assuming the positivity of Yamabe invariant and conformal invariant .
Acknowledgements. The first author is supported in part by the NSF grant DMS–1206661. We would like to thank Professor Alice Chang and Paul Yang for valuable discussions.
2. The method of continuity and openness
Dividing by (recall ) and denoting
equation (1.22) is equivalent to
here and .
To write this equation in terms of the conformal factor , observe that the Schouten tensor of the conformal metric is given by
Using the formula (1.9) we also have
Also, the condition that is equivalent to the inequality
We begin with a fact which permits us to start the continuity process.
Let be a smooth compact Riemannian manifold with dimension . If , then there exists a metric with
we can find close enough to such that . In particular is a solution of (2.4), with
It is clear that . Indeed in this case is a solution. On the other hand if we can show , then it follows there exists a metric with and . To achieve this we will show is both open and closed.
To prove that is open, we consider the linearized operator. To this end, define the map
Then if and only if is a solution of (2.4). Let denote the linearization of at :
To compute we use the standard formulas for the variation of the curvature and the Schouten tensor
In our setting, using
The openness of follows from implicit function theorem and standard elliptic theory, together with the following lemma:
Let be a smooth compact Riemannian manifold with dimension . If
then the operator
is positive definite.
For any smooth function ,
Using the Bochner formula
Note for and , all coefficients before the integral sign are nonnegative. As a consequence is positive definite.
For , note that the coefficient of is equal to and it is negative when is close to .
3. Apriori estimate
Assume is a smooth compact Riemannian manifold with dimension . If , and , then any smooth positive solution to (2.4) with satisfies
Here is independent of and .