Prescribed Q -curvature flow on closed manifolds of even dimension
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Calculus of Variations
Prescribed Q-curvature flow on closed manifolds of even dimension ´ˆ Anh Ngô1,2 · Hong Zhang3,4 Quoc Received: 18 March 2018 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract On a closed Riemannian manifold (M, g0 ) of even dimension n ≥ 4, the well-known prescribed Q-curvature problem asks whether there is a metric g comformal to g0 such that its Q-curvature, associated with the GJMS operator Pg , is equal to a given function f . Letting g = e2u g0 , this problem is equivalent to solving Pg0 u + Q g0 = f enu , where Q g0 denotes the Q-curvature of g0 . The primary objective of the paper is to introduce the following negative gradient flow of the time dependent metric g(t) conformal to g0 , f Q g(t) dμg(t) ∂ g(t) f g(t) = −2 Q g(t) − M 2 ∂t M f dμg(t)
for t > 0,
to study the problem of prescribing Q-curvature. Since M Q g dμg is conformally invariant, our analysis depends on the size of M Q g0 dμg0 which is assumed to satisfy M
Q 0 dμg0 = k(n − 1)! vol(Sn ) for all k = 2, 3, . . .
The paper is twofold. First, we identify suitable conditions on f such that the gradient flow defined as above is defined to all time and convergent, as time goes to infinity, sequentially or uniformly. Second, we show that various existence results for prescribed Q-curvature problem can be derived from the convergence of the flow. Keywords Q-curvature · Negative gradient flow · Closed manifolds · Even dimension Mathematics Subject Classification Primary 53C44; Secondary 35J30
Communicated by A. Chang. Extended author information available on the last page of the article 0123456789().: V,-vol
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Q. A. Ngô, H. Zhang
1 Introduction On a closed manifold (M, g) of dimension n ≥ 3, a formally self-adjoint geometric differential operator A g of the metric g is called conformally covariant of bidegree (a, b) if A gw (ϕ) = e−bw A g (eaw ϕ)
(1.1)
for all ϕ ∈ C ∞ (M), where gw := e2w g is a conformal metric to g. A typical geometric differential operator in conformal geometry is the second-order conformal Laplacian which is defined by L g (ϕ) := −g ϕ +
n−2 Rg ϕ, 4(n − 1)
where Rg is the scalar curvature of g. A well-known fact is that L g is conformally covariant of bidegree ((n − 2)/2, (n + 2)/2) in the sense of (1.1) since L gw (ϕ) = e−
(n+2)w 2
L g (e
(n−2)w 2
ϕ), ∀ ϕ ∈ C ∞ (M).
(1.2)
If we write u 4/(n−2) = e2w , then Eq. (1.2) is changed into n+2
L gw (ϕ) = u − n−2 L g (uϕ), ∀ ϕ ∈ C ∞ (M). By setting ϕ ≡ 1, we get, with the conformal metric gu = u 4/(n−2) g, the transformation law of the scalar curvature Rg −
n+2 4(n − 1) g u + Rg u = Rgu u n−2 . n−2
(1.3)
In the literature, Eq. (1.3) is closely related to the prescribed scalar curvature problem which is to find a positive smooth function u, for a given function f , such that Rgu = f . This challenging problem has already captured much attention by many mathematicians during the past few decades. In the case that the function f is constant, the prescribed scalar cur
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