CR Nirenberg problem and zero Wester scalar curvature
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CR Nirenberg problem and zero Wester scalar curvature Pak Tung Ho1,2 · Seongtag Kim3 Received: 30 January 2020 / Accepted: 5 June 2020 © Springer Nature B.V. 2020
Abstract In this paper, we study the prescribing Webster scalar curvature problem on strictly pseu‑ doconvex CR manifolds of real dimension 2n + 1 . First, we study the CR Nirenberg prob‑ lem and prove some existence results. Second, we provide the existence and a positive lower bound for a solution of the CR Yamabe problem with zero Webster scalar curvature on noncompact complete manifolds. Keywords CR manifold · CR Yamabe problem · Webster scalar curvature · CR Nirenberg problem Mathematics Subject Classification Primary 32V05 · Secondary 53C21
1 Introduction Let (M, 𝜃) be a compact pseudoconvex manifold of real dimension 2n + 1 with the contact 𝜃̃ conformal to 𝜃 such that its form 𝜃 . The CR Yamabe problem is to find a contact form 2 Webster scalar curvature is constant. If we write 𝜃̃ = u n 𝜃 for some 0 < u ∈ C∞ (M) , then their Webster scalar curvatures R𝜃 and R𝜃̃ are related by ) ( 2 2 Δ u + R𝜃 u = R𝜃̃ u1+ n . − 2+ (1) n 𝜃 Therefore, the CR Yamabe problem is to find positive smooth function u in M satisfying (1) with R𝜃̃ being constant. The CR Yamabe problem has been solved in [4, 5, 9, 10, 19–21]. Now we suppose that (M, 𝜃) is a noncompact strictly pseudoconvex CR manifold of real dimension 2n + 1 . One can still consider the CR Yamabe problem on (M, 𝜃) , i.e. the noncompact CR Yamabe problem. In [16], the authors have proved an existence result
* Seongtag Kim [email protected] Pak Tung Ho [email protected] 1
Department of Mathematics, Sogang University, Seoul 04107, South Korea
2
Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun‑gu, Seoul 02455, South Korea
3
Department of Mathematics, Inha University, Incheon 22212, South Korea
13
Vol.:(0123456789)
Annals of Global Analysis and Geometry
when the CR Yamabe constant of (M, 𝜃) is positive. In Sect. 3, by assuming that the Webster scalar curvature R𝜃 is closed to zero, we will find a conformal contact form 2 𝜃̃ = u n 𝜃 such that its Webster scalar curvature R𝜃̃ is zero. In view of (1), it is equivalent to finding a positive smooth solution u to the following equation:
−Δ𝜃 u +
n R u = 0, 2n + 2 𝜃
(2)
To state our theorem, we define the following CR Yamabe invariant:
Y(M, 𝜃) =
u∈C0∞ (M) ∫M
inf
) n /( ) n+1 n 2+ n2 2 R𝜃 u dV𝜃 u dV𝜃 . |∇𝜃 u| + ∫ 2n + 2 M
(
2
By applying similar techniques in [22], we will prove the following theorem in Sects. 3 and 4.
Theorem 1 Let (M, 𝜃) be a noncompact strictly pseudoconvex CR manifold of real dimension 2n + 1 with Webster scalar curvature R𝜃 and infinite volume. Assume that Y(M, 𝜃) > 0
|R |(2n+2)∕(n+2) + |R𝜃 |n+1 dV𝜃 < ∞ . Then there exists a conformal contact form ∫2M 𝜃 𝜃̃ = u n 𝜃 with zero Webster scalar curvature such that ( )n∕(2n+2) |∇𝜃 (u − 1)|2 dV𝜃 + |u − 1|(2n+2)∕n dV𝜃 ∫M ∫M [( )(n+2)∕(n+1) ( )(n+2)∕(2n+2) ] (3) (2n+2)∕(n+2) (2n+2)∕(n+2) 0. − 2+ n
(4)
This problem was considered in [6, 7, 13–15, 26, 28, 29]. I
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