Conformal metrics on $\mathbb R^{2 m}$ with constant $Q$-curvature and large volume

Abstract

We study conformal metrics g u = e 2u |dx| 2 on R 2m with constant Q-curvature Q gu ≡ (2m − 1)! (notice that (2m − 1)! is the Q-curvature of S 2m ) and finite volume. When m = 3 we show that there exists V * such that for any V ∈ [V * , ∞) there is a conformal metric g u = e 2u |dx| 2 on R 6 with Q gu ≡ 5! and vol(g u ) = V . This is in sharp contrast with the fourdimensional case, treated by C-S. Lin. We also prove that when m is odd and greater than 1, there is a constant V m > vol(S 2m ) such that for every V ∈ (0, V m ] there is a conformal metric g u = e 2u |dx| 2 on R 2m with Q gu ≡ (2m − 1)!, vol(g) = V . This extends a result of A. Chang and W-X. Chen. When m is even we prove a similar result for conformal metrics of negative Q-curvature.

Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

“…The above result, which also holds in odd dimension 5 and higher, completely solves problems left open in [14,22,10], proving that in dimension 5 and higher one can find conformally Euclidean metrics with constant Q-curvature and total Q-curvature arbitrarily large, in contrast to the 4 dimensional case, where the total Q-curvature can be at most Λ 1 = 6|S 4 | as shown by [15]. Theorem 1.2 is naturally related to Theorem E because in case iv) of Theorem 1.1, an amount Λ 1 of Q-curvature concentrates at the origin, and we expect to have additional curvature concentrating at S ϕ .…”

“…building upon [17]. A sequence (u k ) of positive (hence radial, by the moving-plane technique) solutions to (22) for some λ k > 0, with u k (0) → ∞ satisfies…”

“…Such solutions exist for every Λ > 0, thanks to [5,10,11,22]. Take u k (x) := u( x k ) − log k, which is also a solution to (77).…”

Conformally Euclidean metrics on ℝn with arbitrary total Q-curvature

“…Although the assumption κ ∈ (0, Λ 1 ] is a necessary condition for the existence of solution to (2) for n = 3 and 4, it is possible to have a solution for κ > Λ 1 arbitrarily large in higher dimension as shown by Martinazzi [18] for n = 6. Huang-Ye [9] extended Martinazzi's result in arbitrary even dimension n of the form n = 4m + 2 for some m ≥ 1, proving that for every κ ∈ (0, ∞) there exists a solution to (2).…”

“…The ideas in [18,9] are based upon ODE theory. One considers only radial solutions so that the equation in (2) becomes an ODE, and the result is obtained by choosing suitable initial conditions and letting one of the parameters go to +∞ (or −∞).…”