As we have already discussed once that a natural conformally invariant in dimension four is the following
which is commonly refered to the Q-curvature of metric , see this topic. Note that, under a conformal change of the metric , the quantity transforms according to
where denotes the Paneitz operator with respect to . Keep in mind that the Paneitz operator is conformally invariant in the sense that
for any conformal metric . For any , the operator acts on a smooth function u on M via the following rule
which plays a similar role as the Laplace operator in dimension two. Observe that , therefore, a simple calculation shows
Hence the total integral is conformally invariant.
Here we have already used the fact that since, by the divergence theorem, we know that
We now cover the following beautiful result due to Gursky published in 1999 in CMP. Before doing so, let us denote by the following
We also denote by the so-called Yamabe invariant given by
We shall prove
Theorem (Gursky). Let be a smooth compact fout-dimensional Riemannian manifold. If then .
Gursky’s proof is quite nice since it makes use of the subcritical equations similarly to the Yamabe approach.
Proof. First, we let solve the following subcritical equation
for each . Since is continuous from the left and non-decreasing by Aubin’s result, we may choose a sequence such that . Let and . Then the scalar curvature of is given by
If we let denote the trace-free Ricci tensor of , then (and this is the key point) as is a conformal invariant, we have
The trace-free Ricci tensor of a metric is defined to be the Ricci tensor subtracts its trace. Mathematically, it is given by
Using the formula for , we obtain
which immediately implies that
Using the formula for shown above, we can estimate
Taking the limit as , we obtain
By the energy estimate of Aubin, . Thus, if , we get
- Prescribed Q-curvature on 4-manifolds
- The Paneitz operator in any dimension
- Flows for Q-curvature problems