Space CPn Research Articles

A comparison theorem for the mean exit time from a domain in a K�hler manifold

Let M be a Kahler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let ω be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary ∂ω. The main result of this paper is a comparison theorem between the Mean Exit Time function defined on ω and the Mean Exit Time from a geodesic ball of the complex projective space ℂℙ n (λ) which involves a characterization of the geodesic balls among the domain ω. In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of ω, using the Index Lemma for Submanifolds.

An isometric embedding of the complex hyperbolic space in a pseudo-euclidean space and its application to the study of real hypesurfaces

In the last years, the use of an idea of A. Ros, [11], has meant an interesting progress in the study of several families of submanifoids of the complex projective space CPn. This idea essentially consists in considering the firststandard embedding of CPn in a certain Euclidean space RN, and contemplating the submanifoids of CPn in the light of that new embedding. The firststandard embedding has parallel second fundamental form and makes CPn to be a symmetric i?-space in RN. In particular real hypersurfaces of CPn have been analysed under this point of view, [6], [13], and new characterizations of this important class of hypersurfaces have been obtained. In 1986, the second author and S. Montiel, [8], made a systematic study of a certain family of real hypersurfaces of the complex hyperbolic space CHn. In the process of classificationof that family they introduced new examples without parallelin CPn. Therefore if we could get an isometric embedding of CHn in some Euclidean space RN provided of as good geometric properties as those of the firststandard embedding of CPn, we could try to profound in the study of real hypersurfaces in CHn. On the other hand fully immersed complete submanifoids of a Euclidean space with parallel second fundamental form have been totally classifiedby D. Ferus, [3], [4]. As a consequence his result implies that a complete irreducible (as a Riemannian manifold) submanifold which is fullyimmersed in an Euclidean space with parallel second fundamental form is congruent to either an hyperplane or to an irreducible symmetric i?-space immersed by means of its standard embedding. Consequentely we see that there exist no an isometric immersion

HOMOTOPY FORMULAS FOR THE $ \bar\partial$-OPERATOR ONCPnAND THE RADON-PENROSE TRANSFORM

Global integral representations are constructed for differential forms on domains in complex projective space CPn. Consequences of these representations are the following: first, criteria for the solvability of the inhomogeneous Cauchy-Riemann equations on q-pseudoconvex and q-pseudoconcave domains in an algebraic manifold; second, explicit formulas and bounds for solutions of these equations; and third, a description of the kernel and image and an inversion formula for the Radon-Penrose transform of (0,q)-forms on q-linearly concave domains in CPn. Bibliography: 23 titles.

Higher dimensional generalizations of some classical theorems on normal meromorphic functions

Generalizing the notion of normality of a meromorphic function to a holomorphic mapping of the unit ball Bm⊂C m into the complex projective space CPN, the author obtains several necessary and sufficient conditions for a holomorphic mapping f: Bm→CPN to be normal, thus generalizing the classical theorems of Lehto-Virtanen, Lohwater-Pommerenke and Lappan. It is also illustrated that the full generalizations of the theorems of Lohwater-Pommerenke and Lappan to a higher-dimensional domain are not likely to be true.

On some real hypersurfaces of a complex projective space

A principal circle bundle over a real hypersurface of a complex projective space CPn can be regarded as a hypersurface of an odddimensional sphere. From this standpoint we can establish a method to translate conditions imposed on a hypersurface of CPn into those imposed on a hypersurface of S2'+1. Some fundamental relations between the second fundamental tensor of a hypersurface of CPn and that of a hypersurface of S2n+1 are given. Introduction. As is well known a sphere S2n+1 of dimension 2n + 1 is a principal circle bundle over a complex projective space CPn and the Riemannian structure on CPn is given by the submersion ir: S2n+ 1 ~ CPn [4], [7]. This suggests that fundamental properties of a submersion would be applied to the study of real submanifolds of a complex projective space. In fact, H. B. Lawson [2] has made one step in this direction. His idea is to construct a principal circle bundle M2n over a real hypersurface M2n-1 of Cpn in such a way that M2n is a hypersurface of S2n + 1 and then to compare the length of the second fundamental tensors of M2n-1 and M2 n. Thus we can apply theorems on hypersurfaces of S2n+1. In this paper, using Lawson's method, we prove a theorem which characterizes some remarkable classes of real hypersurfaces of Cpn. First of all, in ?1, we state a lemma for a hypersurface of a Riemannian manifold of constant curvature for the later use. In ?2, we recall fundamental formulas of a submersion which are obtained in [4], [7] and those established between the second fundamental tensors of M and M. In ?3, we give some identities which are valid in a real hypersurface of CPn. After these preparations, we show, in ?4, a geometric meaning of the commutativity of the second fundamental tensor of M in Cp'n and a fundamental tensor of the submersion ir: M . M 1. Hypersurfaces of a Riemannian manifold of constant curvature. Let M be an (m + 1)-dimensional Riemannian manifold with a Riemannian metric G and i: M M be an isometric immersion of an m-dimensional differentiable Received by the editors March 25, 1974 and, in revised form, September 9, 1974. AMS (MOS) subject classifications (1970). Primary 53C40, 53C20.