Contact strongly pseudo-convex integrable CR metrics as critical points
Let \((\eta,\xi,\varphi,g)\) be a contact Riemannian structure on a \((2n+1)\)-dimensional differentiable manifold \(M\) [see, e.g., \textit{D. E. Blair} (Lect. Notes in Math. 509, Springer, Berlin) (1976; Zbl 0319.53026)]. \textit{S. Tanno} [Trans. Am. Math. Soc. 314, 349-379 (1989; Zbl 0677.53043] has introduced the notion of strongly pseudo-convex integrable CR-structures. Analytically this structure can be characterized as that contact Riemannian structure which satisfies additionally the integrability condition \[ (\nabla_X\varphi)Y-g(X+hX,Y)\xi+\eta(Y)(X+hX)=0, \] where \(2h={\mathcal L}_{\xi}\varphi\) (\(\mathcal L\) is the Lie derivative operator). In the paper under review, it is proved that the contact strongly pseudo-convex integrable CR-metrics \(g\) on compact manifolds \(M\) are critical points of the functional \(E_1(g)-E_2(g)\), where \(E_{\alpha}(g)=\int_M W_{\alpha} dM\) and \(W_{\alpha}\), \(\alpha=1,2\), are certain generalizations of the so-called Webster curvature [see \textit{S. Tanno}, ibid., and \textit{D. E. Blair} and \textit{D. Perrone}, Can. Math. Bull. 35, 455-462 (1992; Zbl 0756.53014)]. Moreover, such a metric \(g\) is a critical point of \(E_1(g)\) or \(E_2(g)\) if \(g\) is \(K\)-contact (i.e., \({\mathcal L}_{\xi}g=0\)).
scientific article; zbMATH DE number 1053892
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