Prescribed Webster scalar curvature on \(S^{2n+1}\) in the presence of reflection or rotation symmetry (Q293525)

Let \((S^n,g_0)\) be the \(n\)-dimensional sphere with the standard Riemannian metric \(g_0\). The Nirenberg problem is to find a metric conformal to \(g_0\) in \(S^n\) such that its scalar curvature is a given smooth function. This problem was studied by many authors. Recently \textit{M. C. Leung} and \textit{F. Zhou} [Proc. Am. Math. Soc. 142, No. 5, 1607--1619 (2014; Zbl 1284.53040)] proved the existence result for prescribed scalar curvature with symmetry. NEWLINENEWLINENEWLINENEWLINE Now let \((S^{2n+1},\theta_0)\) be the \((2n+1)\)-dimensional sphere with the contact form \(\theta_0=\sqrt {-1}\sum_{i=1}^{n+1}(\zeta_id\bar \zeta_i-\bar\zeta_id\zeta_i)\), where \(S^{2n+1}=\{(\zeta_i,\dots,\zeta_{n+1})\in \mathbb C^{n+1}: |\zeta_1|^2+\cdots+|\zeta_{n+1}|^2=1\}\subset \mathbb C^{n+1}\). One can consider the similar problem to the Nirenberg problem for the compact strictly pseudoconvex CR manifold \((S^{2n+1},\theta_0)\). For a given smooth function \(f\) on \(S^{2n+1}\), find a contact form \(\theta\) conformal to \(\theta_0\) such that its Webster scalar curvature \(R_{\theta}\) equals \(f\).NEWLINENEWLINEMotivated by the paper of Leung and Zhou [loc. cit.], using the Webster scalar curvature flow, which is a generalization of the CR Yamabe flow, the author proves the existence result for prescribed Webster scalar curvature with symmetry.NEWLINENEWLINENEWLINENEWLINEAssumption 1. Let \(f\) be a symmetric function for a mirror reflection upon a hyperplane \(H\subset \mathbb C^{n+1}\) passing through the origin. Without loss of generality one can assume that \(F=\{(0,\zeta_2,\dots,\zeta_{n+1})\in S^{2n+1}\}=H\cap S^{2n+1}\). NEWLINENEWLINENEWLINENEWLINE Assumption 2. Let \(f\) be a function invariant under the rotation \(\gamma_{\phi}\) of angle \(\phi=\frac {\pi}{k}\), \(k>1\), \(k\in \mathbb N\) (with an axis being a straight line in \(\mathbb C^{n+1}\) passing through the origin). Without loss of generality one can assume that the straight line is the \(\zeta_{n+1}\)-axis and \(F=\{(0,\dots,0,\zeta_{n+1})\in S^{2n+1}\}\) is the fixed point set. NEWLINENEWLINENEWLINENEWLINE The main theorem of the paper is the following. Let \(f\) be a smooth positive function on \(S^{2n+1}\) which is invariant under the symmetry described either in Assumption 1 or in Assumption 2. Assume also thatNEWLINENEWLINE(a) if \(x_m\in F\), \(f(x_m)=\max_Ff\), then \(\Delta_{\theta_0}f(x_m)>0\) andNEWLINENEWLINE(b) \(\max_{S^{2n+1}}f<2^{\frac 1n}\max_Ff\).NEWLINENEWLINEThen \(f\) can be realized as the Webster scalar curvature of some contact form conformal to \(\theta_0\).