Showing papers in "Tohoku Mathematical Journal in 1997"

Spacelike hypersurfaces of constant mean curvature and calabi-bernstein type problems

Abstract: Spacelike graphs of constant mean curvature over compact Riemannian manifolds in Lorentzian manifolds with constant sectional curvature are studied. The corresponding Calabi-Bernstein type problems are stated. In the case of nonpositive sectional curvature all their solutions are obtained, and for positive sectional curvature well-known results are extended.

Complex extensors and Lagrangian submanifolds in complex Euclidean spaces

Abstract: Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed curve F in the complex plane, we introduce the notion of the complex extensor of M in the complex Euclidean «-space via F. The main purpose of this paper is to classify Lagrangian //-umbilical submanifolds of the complex Euclidean «-space by utilizing complex extensors. We prove that, except the flat ones, Lagrangian //-umbilical submanifolds of complex Euclidean «-space with n greater than 2 are Lagrangian pseudo-spheres and complex extensors of the unit hypersphere of the Euclidean w-space. For completeness we also include in the last section the classification of flat Lagrangian //-umbilical submanifolds of complex Euclidean spaces.

The structure of the polytope algebra

Abstract: We construct an isomorphism from McMullen's polytope algebra, onto the quotient of the algebra of continuous, piecewise polynomial functions with integral value at 0, by its ideal generated by coordinate functions. This explains the non-trivial grading of the polytope algebra, by the obvious grading of piecewise polynomial functions. In the process of the proof, we make explicit many connections between convex poly topes and piecewise polynomials. Introduction. In the study of valuations (or finitely additive measures) on convex polytopes in a finite-dimensional real vector space, a fundamental role is played by the polytope algebra: the universal group for translation-invariant valuations. This group is endowed with a multiplication, via Minkowski sum of polytopes, and with many other structures, discovered by McMullen, Morelli, Khovanskii-Pukhlikov and others. In particular, the polytope algebra is almost a graded algebra over R; its grading is defined by diagonalizing the action of the group of dilatations (see [Mel]). The proof of existence of this grading uses the logarithm of a polytope P, defined by log(P) = ΣTM=1(—iY~ (P—'ίY/n (this makes sense in the polytope algebra, because P — 1 is nilpotent there). In this paper, we recover some of the most important properties of the polytope algebra, as corollaries of a structure theorem for this algebra. To state our main result, we need some notation. Let V be a vector space over R of finite dimension d>2, and let V* be its dual. To any convex polytope P in V* is associated its support function HP on V; then HP is continuous, and piecewise linear with respect to some subdivision of Finto polyhedral cones having the origin as their common vertex. We denote by R the algebra of all continuous functions on V that are piecewise polynomial (in the same sense). Then R is a graded algebra over R for the operations of pointwise addition and multiplication; it turns out that R is generated by support functions of polytopes. We denote by R the quotient of R by its graded ideal generated by all (globally) linear functions on V. THEOREM, (i) The graded algebra R=(BTM=0Rn vanishes in all degrees n>d. Moreover, the vector space Rd is one-dimensional, and multiplication in R induces non-degenerate pairings Rj x Rd_j-> Rd for l

Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus

Abstract: In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same symmetry properties exists for all given minimal surfaces satisfying certain conditions. The surfaces we construct in this paper are irreducible, and in the process of showing this, we also prove some results about the reducibility of surfaces. Furthermore, in the case that the surfaces are of genus 0, we are able to make some estimates on the range of the parameter for the one-parameter family.

Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems

Abstract: We consider here a general Lotka-Volterra type ^-dimensional periodic functional differential system. Sufficient conditions for the existence, uniqueness and global asymptotic stability of periodic solutions are established by combining the theory of monotone flow generated by FDEs, Horn's asymptotic fixed point theorem and linearized stability analysis. These conditions improve and generalize the recent ones obtained by Freedman and Wu (1992) for scalar equations. We also present a nontrivial application of our results to a delayed nonautonomous predator-prey system.

On the Iwasawa invariants of certain real abelian fields

Abstract: For any totally real number field k and any prime number /?, the Iwasawa lambda-invariant and the mu-invariant are conjectured to be both zero. We give a new efficient method to verify this conjecture for certain real abelian fields. The new features of our method compared with other existing ones are that we use effectively cyclotomic units and that we introduce a new way to apply p-aάic L-functions to the conjecture.

A characterization of the mean curvature functions of codimension-one foliations

Abstract: Walczak posed a problem on the characterization of the mean curvature functions of codimension-one foliations. An affirmative answer to this problem is given here. As an application, we get a simpler proof of the topological characterization, due to the author, of codimension-one foliations consisting of constant mean curvature hypersurfaces.

An integral representation of eigenfunctions for Macdonald's q-difference operators

Abstract: We give the eigenfunctions for Macdonald's q-diierence operators in terms of q-Selberg type integrals. Our result can be applied not only to the case of Macdonald symmetric polynomials but also to the cases of rational and meromorphic solutions.

Existence of almost periodic solutions of some functional-differential equations with infinite delay in a Banach space

Abstract: For functional differential equations with infinite delay in a Banach space, the existence of almost periodic solutions is studied under some stability assumptions.

Pointwise convergence of Fejer type means

Abstract: Let P be a compact n-dimensional convex polyhedron in R n containing the origin in its interior and let e H(t) = Z 1 0 Z vP e 2it ddv, t2 R n ;where vP is the characteristic function of the dilated polyhedron vP. Let H N (t) = X m2Z n e H 1 N+1 (t+m), t 2 T n , where e H " (t) = " ?n e H(t="). We prove that (e H " f)(t) ! f(t) a.e., as " ! 0, for any f 2 L 1 (R n), and that (H N f)(t) ! f(t) a.e., as N ! 1, for any f 2 L 1 (T n).

Remarks on a conjecture of Batyrev and Manin

Abstract: We give several remarks on a conjecture of Batyrev and Manin on the distribution of rational points on algebraic varieties. We study the signature of the geometric invariant of Batyrev-Manin in relation to the value of the Kodaira dimension, and study the validity of the Batyrev-Manin conjecture under the assumption that there exists an unramified covering for which the Batyrev-Manin conjecture holds.