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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 n2 and a real number c, a complex space form M n (c) 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 (c). 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 (c). 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 (c).
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
35B45A priori estimates for solutions of PDE
35J60Nonlinear elliptic equations
53A30Conformal differential geometry