Showing papers in "Tohoku Mathematical Journal in 1991"

Notes on canonical surfaces

Abstract: A minimal algebraic surface S is called a canonical surface if the map Φκ: S-+P , n=pg—\\, associated to the canonical system \\K\\ induces a birational map of S onto its image. Let Quad(S) denote the intersection of all the quadrics through the image ΦK(S). If S is a canonical surface, then c\\>2>pg — l (see [10, Part II, Lemma 1.1]). If the equality sign holds here, S has rather simple structure and its construction can be completely described as in [2]. These are all essentially due to Castelnuovo [4], and I obtained my proof in 1976, which is mostly similar to [2, §§1-4]. Moreover, I noticed that some of the canonical surfaces with pg = l, c\\ = \\<\\ (such that Quad(S) is a cone over the Veronese surface) have obstructed deformations. For such S, | K | is not ample, and the canonical system | Kt | remains non-ample for any small deformation St of S. So, by [3], S has generically non-reduced moduli. This was insinuated in [10, Part III, Remark on p. 229], but with an erroneous citation pg — β, c\\ = 11. (I planned to write a paper entitled \"On certain canonical surfaces\" to discuss surfaces with c\\ = 3pg — l and 3pg — β, but it was never completed.) This surface was independently found recently by Miranda [15]. But he missed one point: If {St: teM} is a flat family over a parameter space M, then does teM} form a flat family? This is not true in general, because the dimension of may jump in some case.

The ideal boundaries of complete open surfaces

Abstract: 0. Introduction. It is an interesting problem to investigate compactiίications of complete noncompact Riemannian manifolds. The ideal boundary of an Hadamard manifold Zis defined to be the set of equivalence classes of rays in X. Here the equivalence relation between two rays in X is obtained by an asymptotic relation between them. Busemann first defined (see [Bu, Chap. 3, §22]) an asymptotic relation between two rays (which he called co-ray relation and used to define parallelism on a straight G-space). This asymptotic relation is not symmetric in general and hence the equivalence classes of rays are not defined by it. If X is an Hadamard manifold, then this asymptotic relation becomes symmetric, and makes it possible to define the ideal boundary X(co) of X (see [EO] and [BGS]). Gromov defined in [BGS] the Tits metric on X{ao). Recently, Kause constructed an ideal boundary of an asymptotically nonnegatively curved manifold. The purpose of the present paper is, first of all, to define the ideal boundary M(oo) with the metric d^ as the set of natural equivalence classes of rays in a finitely connected, oriented, complete and noncompact surface M admitting total curvature. Then we investigate the geometry on the ideal boundary in terms of the total curvature. Here the total curvature c(M) of such a surface M is defined by an improper integral over M of the Gaussian curvature G:

Qualitative analysis of a nonautonomous nonlinear delay differential equation

Abstract: This paper is devoted to the systematic study of some qualitative properties of solutions of a nonautonomous nonlinear delay equation, which can be utilized to model single population growths. Various results on the boundedness and oscillatory behavior of solutions are presented. A detailed analysis of the global existence of periodic solutions for the corresponding autonomous nonlinear delay equation is given. Moreover, sufficient conditions are obtained for the solutions to tend to the unique positive equilibrium.

The rigidity for real hypersurfaces in a complex projective space

Abstract: The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.