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Prescribing a fourth order conformal invariant on the standard sphere. II. Blow up analysis and applications. (English) Zbl 1150.53012
Given an integer $n\ge 2$ and a real number $c$, a complex space form ${M}_{n}\left(c\right)$ is a complex $n$-dimensional Kaehler manifold of constant holomorphic curvature $c$. In 1986, considering an appropriate integrable $2n-2$-dimensional distribution $D$, M. Kimura [Trans. Am. Math. Soc. 296, 137–149 (1986; Zbl 0597.53021)] gave a construction of a ruled real hypersurface $M$ which is foliated by Einstein complex hypersurfaces in ${M}_{n}\left(c\right)$. Such a manifold $M$ with the given distribution $D$ is called an Einstein ruled real hypersurface. In the paper under review, the author generalizes that notion by introducing a pseudo-Einstein ruled real hypersurface defined as a ruled real hypersurface $M$ which is foliated by pseudo-Einstein complex hypersurfaces of ${M}_{n}\left(c\right)$. Some examples illustrate his definition. Then, he gives a new characterization of this kind of pseudo-Einstein ruled real hypersurfaces in terms of the Ricci tensor and some integrability condition defined on a distribution orthogonal to the structure vector field in ${M}_{n}\left(c\right)$.
##### MSC:
 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (global) 35B45 A priori estimates for solutions of PDE 35J60 Nonlinear elliptic equations 53A30 Conformal differential geometry