Showing papers in "Tohoku Mathematical Journal in 2005"
TL;DR: In this paper, the authors prove an integral maximum principle for random walks on graphs, and give several applications to pointwise estimates of their transition probabilities, including the time-dependent case.
Abstract: We prove an integral maximum principle for random walks on graphs, and give several applications to pointwise estimates of their transition probabilities, including the time-dependent case.
53 citations
TL;DR: In this paper, two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold with parallel Lee form have been studied, and the Lee-Cauchy-Riemann (LCR) transformations have been introduced as a class of diffeomorphisms preserving the specific manifold structure.
Abstract: We study two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold. First, we study compact l.c.K. manifolds by means of the existence of holomorphic l.c.K. flow (i.e., a conformal, holomorphic flow with respect to the Hermitian metric.) We characterize the structure of the compact l.c.K. manifolds with parallel Lee form. Next, we introduce the Lee-Cauchy-Riemann ($\mathrm{LCR}$) transformations as a class of diffeomorphisms preserving the specific $G$-structure of l.c.K. manifolds. We show that compact l.c.K. manifolds with parallel Lee form admitting a non-compact holomorphic flow of $\mathrm{LCR}$ transformations are rigid: such a manifold is holomorphically isometric to a Hopf manifold with parallel Lee form.
39 citations
TL;DR: In this paper, smooth compact complex surfaces admitting non-trivial surjective endomorphisms are classified up to isomorphism, and the following surfaces are listed in the non-algebraic case: a complex torus, a Kodaira surface, a Hopf surface with at least two curves, a successive blowups of an Inoue surface with curves whose centers are nodes of curves, and an inoue surfaces without curves satisfying a rationality condition.
Abstract: Smooth compact complex surfaces admitting non-trivial surjective endomorphisms are classified up to isomorphism. The algebraic case was dealt with earlier by the authors. The following surfaces are listed in the non-algebraic case: a complex torus, a Kodaira surface, a Hopf surface with at least two curves, a successive blowups of an Inoue surface with curves whose centers are nodes of curves, and an Inoue surface without curves satisfying a rationality condition.
32 citations
31 citations
TL;DR: In this article, the authors established an analogue of Beurling's uncertainty principle for the group Fourier transform on the Euclidean motion group and proved the most general version of Hardy's theorem on it, which characterises functions on the motion group that are controlled by the heat kernel associated to the Laplacian of the space.
Abstract: We establish an analogue of Beurling's uncertainty principle for the group Fourier transform on the Euclidean motion group We also prove the most general version of Hardy's theorem on it which characterises functions on the motion group that are controlled by the heat kernel associated to the Laplacian of the Euclidean space
21 citations
TL;DR: In this article, an algebraic theory of ind-schemes that allows us to represent finite $K$ schemes as infinite dimensional lattices of fixed discriminant in the vector space is presented.
Abstract: Let $k$ denote the algebraic closure of the finite field, $\mathbb F_p,$ let $\mathcal O$ denote the Witt vectors of $k$ and let $K$ denote the fraction field of this ring. In the first part of this paper we construct an algebraic theory of ind-schemes that allows us to represent finite $K$ schemes as infinite dimensional $k$-schemes and we apply this to semisimple groups. In the second part we construct spaces of lattices of fixed discriminant in the vector space $K^n.$ We determine the structure of these schemes. We devote particular attention to lattices of fixed discriminant in the lattice, $p^{-r}\mathcal O^n,$ computing the Zariski tangent space to a lattice in this scheme and determining the singular points.
18 citations
TL;DR: In this paper, the concepts of rounding and flattening of a smooth map $g$ of an $m$-dimensional manifold $M$ to the euclidean space (R^n$ with $m < n) were introduced.
Abstract: We introduce the concepts of rounding and flattening of a smooth map $g$ of an $m$-dimensional manifold $M$ to the euclidean space $\R^n$ with $m
16 citations
TL;DR: In this article, the authors characterized the invariant structures of (1, 2)-symplectic metrics on the maximal flag manifold and showed that these structures are invariant for almost complex structures.
Abstract: We characterize the invariant $f$-structures $\mathcal{F}$ on the classical maximal flag manifold $\mathbb F(n)$ which admit (1,2)-symplectic metrics. This provides a sufficient condition for the existence of $\mathcal{F}$-harmonic maps from any cosymplectic Riemannian manifold onto $\mathbb F(n)$. In the special case of almost complex structures, our analysis extends and unifies two previous approaches: a paper of Brouwer in 1980 on locally transitive digraphs, involving unpublished work by Cameron; and work by Mo, Paredes, Negreiros, Cohen and San Martin on cone-free digraphs. We also discuss the construction of (1,2)-symplectic metrics and calculate their dimension. Our approach is graph theoretic.
14 citations
TL;DR: In this article, the authors discuss the local triviality in the ideal class group of the basic Z_p$-extension over an imaginary quadratic field and prove that such triviality distributes with natural density.
Abstract: We shall discuss the local triviality in the ideal class group of the basic $\mathbf Z_p$-extension over an imaginary quadratic field and prove, in particular, a result which implies that such triviality distributes with natural density $1$.
14 citations
TL;DR: In this article, the Hardy space H 1 F (R n ) associated with a family F of sections and a doubling measure µ is defined, which is closely related to the Monge-Ampere equation.
Abstract: In this paper we define the Hardy space H 1 F (R n ) associated with a family F of sections and a doubling measure µ ,w hereF is closely related to the Monge-Ampere equation. Furthermore, we show that the dual space of H 1 F (R n) is just the space BMOF (Rn), which was first defined by Caffarelli and Gu tierrez. We also prove that the Monge-Ampere singular integral operator is bounded from H 1 F (R n ) to L 1 (R n ,dµ ). 1. Introduction. In 1996, Caffarelli and Gutierrez (CG1) studied real variable theory related to the Monge-Ampere equation. They gave a Besicovitch type covering lemma for a family F of convex sets in Euclidean n-space R n ,w hereF ={ S(x, t); x ∈ R n and t> 0} and S(x, t) is called a section (see the definition below) satisfying certain axioms of affine invariance. In terms of the sections, Caffarelli and Gutierrez set up a variant of the Calderon- Zygmund decomposition by applying this covering lemma and the doubling condition of a Borel measure µ. The decomposition plays an important role in the study of the linearized Monge-Ampere equation (CG2). As an application of the above decomposition, Caffarelli and Gutierrez defined the Hardy-Littlewood maximal operator M and BMOF (R n ) space as- sociated to a family F of sections and the doubling measure µ, and obtained the weak type (1,1) boundedness of M and the John-Nirenberg inequality for BMOF (R n ) in (CG1).
14 citations
TL;DR: In this article, the authors investigated higher dimensional analogues for open submanifolds of Euclidean space with cone-like ends and showed that the curvature defect can be positive, zero, or negative depending on the shape of the ends "at infinity''.
Abstract: The classical Cohn-Vossen inequality states that for any complete 2-dimensional Riemannian manifold the difference between the Euler characteristic and the normalized total Gaussian curvature is always nonnegative. For complete open surfaces in Euclidean 3-space this curvature defect can be interpreted in terms of the length of the curve "at infinity''. The goal of this paper is to investigate higher dimensional analogues for open submanifolds of Euclidean space with cone-like ends. This is based on the extrinsic Gauss-Bonnet formula for compact submanifolds with boundary and its extension "to infinity''. It turns out that the curvature defect can be positive, zero, or negative, depending on the shape of the ends "at infinity''. We give an explicit example of a 4-dimensional hypersurface in Euclidean 5-space where the curvature defect is negative, so that the direct analogue of the Cohn-Vossen inequality does not hold. Furthermore we study the variational problem for the total curvature of hypersurfaces where the ends are not fixed. It turns out that for open hypersurfaces with cone-like ends the total curvature is stationary if and only if each end has vanishing Gauss-Kronecker curvature in the sphere "at infinity''. For this case of stationary total curvature we prove a result on the quantization of the total curvature.
TL;DR: Bressler and Lunts as mentioned in this paper gave a straightforward approach to combinatorial duality and a natural intersection product, completely within the framework of elementary sheaf theory and commutative algebra, thus avoiding derived categories.
Abstract: The proof of the Combinatorial Hard Lefschetz Theorem for the "virtual'' intersection cohomology of a not necessarily rational polytopal fan as presented by Karu completely establishes Stanley's conjectures for the generalized $h$-vector of an arbitrary polytope. The main ingredients, Poincare Duality and the Hard Lefschetz Theorem, rely on an intersection product. In its original constructions, given independently by Bressler and Lunts on the one hand, and by the authors of the present article on the other, there remained an apparent ambiguity. The recent solution of this problem by Bressler and Lunts uses the formalism of derived categories. The present article instead gives a straightforward approach to combinatorial duality and a natural intersection product, completely within the framework of elementary sheaf theory and commutative algebra, thus avoiding derived categories.
Abstract: We show that under certain symmetry, the images of complete harmonic embeddings from the complex plane into the hyperbolic plane is completely determined by the geometric information of the vertical measured foliation and is independent of the horizontal measured foliation of the corresponding Hopf differentials
TL;DR: In this article, the authors investigate the dynamics of a broader class of functions which are meromorphic outside a compact totally disconnected set and establish the connections between the Fatou components and the singularities of the inverse function.
Abstract: In this paper, we investigate the dynamics of a broader class of functions which are meromorphic outside a compact totally disconnected set. We shall establish the connections between the Fatou components and the singularities of the inverse function and, accordingly, give sufficient conditions for the non-existence of wandering domains or Baker domains, and for the Julia set to be the Riemann sphere. Through the discussion of permutability of such functions, we shall construct several transcendental meromorphic functions which have Baker domains and wandering domains with special properties; for example, wandering and Baker domains with a critical value on the boundary and a wandering domain with the boundary being a Jordan curve (some such examples for entire functions were exhibited in other papers) and those of non-finite type which have no wandering domains.
TL;DR: In this article, characterizations of invertible and Fredholm composition operators for Bergman spaces on connected domains in the complex plane have been obtained and the isomorphism between Toeplitz algebras and their K-theory is discussed.
Abstract: Characterizations of invertible and Fredholm composition operators are obtained for Bergman spaces on connected domains in the complex plane. In addition, the isomorphism between Toeplitz algebras and their K-theory are discussed.
TL;DR: In this paper, the authors classify real hypersurfaces all of whose geodesics orthogo-nal to the characteristic vector field are plane curves in complex projective or complex hyper-bolic spaces.
Abstract: In this paper we classify real hypersurfaces all of whose geodesics orthogo- nal to the characteristic vector field are plane curves in complex projective or complex hyper- bolic spaces.
TL;DR: In this article, the transplantation operators for the Hankel transform were considered and their boundedness on the real Hardy space was established, and the Hormander-Mihlin type multiplier theorem for the transplant operator was obtained.
Abstract: The transplantation operators for the Hankel transform are considered and their boundedness on the real Hardy space is established. As its application, we obtain the Hormander-Mihlin type multiplier theorem for the Hankel transform on the real Hardy space.
TL;DR: In this paper, an analogue of the Dieudonne module is constructed and an equivalence between the categories of formal groups and right modules satisfying certain con- ditions is established.
Abstract: Honda's theory gives an explicit description up to strict isomorphism of for- mal groups over perfect fields of characteristic p �= 0 and over their rings of Witt vectors by means of attaching a certain matrix, which is called its type, to every formal group. A dual no- tion of right type connected with the reduction of the formal group is introduced while Honda's original type becomes a left type. An analogue of the Dieudonne module is constructed and an equivalence between the categories of formal groups and right modules satisfying certain con- ditions, similar to the classical anti-equivalence between the categories of formal groups, and left modules satisfying certain conditions is established. As an application, the � -isomorphism classes of the deformations of a formal group over and the action of its automorphism group on these classes are studied. 0. Introduction. Let k be a perfect field of characteristic p �= 0a ndO its ring of Witt vectors. Honda's theory (5) gives an explicit description of formal groups over O and k. It attaches to every n-dimensional formal group F a certain n × n-matrix over the non- commutative twisted power series ring E, which is called a type of F. If we restrict our consideration to the p-typical formal groups we will not lose much: every formal group under consideration is strongly isomorphic to a p-typical group. In the present paper, we attach to every p-typical formal group another matrix over E, which we call a 'right type', while Honda's original type will be a 'left type'. The left type describes formal groups up to strict isomorphism and the right type is connected with the reduction of formal groups. In a sense, these notions are dual. Fontaine (4) used Honda's technique to construct an anti-equivalence from the categories of formal groups over k and that over O to the category of left E-modules and the category of the pairs consisting of a left E-module and its O-submodule satisfying certain conditions, respectively. Moreover, the reduction modulo p functor between the former categories corre- sponds to the forgetting of the second component functor between the latter ones. Employing the notion of right type, we obtain an equivalence from the categories of the formal groups over k and that over O to the category of right E-modules and the category of the pairs con- sisting of a right E-module and its O-submodule satisfying certain conditions, respectively. In this construction, the reduction modulo p functor corresponds to the functor of factorization of the first component by the E-linear envelope of the second component.
TL;DR: In this article, the pullback of an eigenform of the Laplacian on the base of a compact Riemannian $V$-submersion is studied.
Abstract: We study when the pull-back of an eigenform of the Laplacian on the base of a compact Riemannian $V$-submersion is an eigenform of the Laplacian on the total space of the submersion, and when the associated eigenvalue can change.
TL;DR: In this paper, a quasiconformal automorphism of a Riemann surface is considered, which fixes the homotopy class of a simple closed geodesic.
Abstract: We consider a quasiconformal automorphism of a Riemann surface, which fixes the homotopy class of a simple closed geodesic. Under certain conditions on the injectivity radius of the surface and bounds on the dilatation of the map, the automorphism induces a periodic element of the Teichmuller modular group. We may also estimate the order of the period.
TL;DR: In this article, the notion of numerically flat vector bundles over a Kahler manifold was generalized to a semisimple complex algebraic group, and it was proved that a vector bundle over a manifold can be considered to be numerically fixed if and only if the corresponding vector bundle (text{ad}(E_G) is numerically flattish.
Abstract: Generalizing the notion of a numerically flat vector bundle over a Kahler manifold $M$, we define a numerically flat principal $G$-bundle over $M$, where $G$ is a semisimple complex algebraic group. It is proved that a principal $G$-bundle $E_G$ is numerically flat if and only if $\text{ad}(E_G)$ is numerically flat. Numerically flat bundles are also characterized using the notion of semistability.
TL;DR: In this article, a complete classification of immersions of parallel Kahler submanifolds (M 2m,J1) in a quaternionic Kahler symmetric space ( f M 4n,e) of non zero scalar curvature was given.
Abstract: The non totally geodesic parallel Kahler submanifolds (M 2n ,J1) of the quaternionic space HP n were classified by K. Tsukada, (Tsu2). Here we give the complete classification of non totally geodesic immersions of parallel Kahler submanifolds (M 2m ,J1) in a quaternionic Kahler symmetric space ( f M 4n ,e) of non zero scalar curvature, i.e. in a Wolf space W or in its non compact dual. They are exhausted by parallel Kahler submanifolds of a totally geodesic submanifold M which is either an Hermitian symmetric space or a quaternionic projective space.
TL;DR: In this article, it was shown that the parabolicity of a manifold is equivalent to the validity of divergence theorem for some class of subharmonic functions, and the Liouville property of harmonic maps on parabolic manifolds.
Abstract: We show that the parabolicity of a manifold is equivalent to the validity of the 'divergence theorem' for some class of $\delta$-subharmonic functions. From this property we can show a certain Liouville property of harmonic maps on parabolic manifolds. Elementary stochastic calculus is used as a main tool.
TL;DR: Fujimoto as mentioned in this paper extended Nevanlinna's uniqueness theorem to the case of moving hyperplanes in pointwise general position, and extended it to nonconstant holomorphic curves.
Abstract: R. Nevanlinna showed, in 1926, that for two nonconstant meromorphic functions on the complex plane, if they have the same inverse images counting multiplicities for four distinct complex values, then they coincide up to a Mobius transformation, and if they have the same inverse images counting multiplicities for five distinct complex values, then they are identical. H. Fujimoto, in 1975, extended Nevanlinna’s result to nondegenerate holomorphic curves. This paper extends Fujimoto’s uniqueness theorem to the case of moving hyperplanes in pointwise general position.
TL;DR: In this article, the behavior of trajectories of the Lotka-Volterra competition system with periodic coefficients under telegraph noise was studied and sufficient conditions for the average permanence were given.
Abstract: In this paper we study the behavior of trajectories of the Lotka-Volterra com- petition system with periodic coefficients under t elegraph noise. We give sufficient conditions for the average permanence. Furthermore, we determine the ω-limit sets of the system. 1. Introduction. In this paper we study the behavior of trajectories of the Lotka- Volterra competition system with periodic coefficients under telegraph noise. Until now, many models have revealed the effect of environmental variability on the population dynamics in mathematical ecology (10, 14). In particular, a great effort has been made to find the possibil- ity of the coexistence of competing species under the unpredictable or rather predictable (such as seasonal) environmental fluctuations. It is well recognized that the seasonality has similar effects to stochastic variation (4, 9), but as Lo reau (11) pointed out, the theory of coexistence in a seasonal environment needs further development to reveal the variety of possibilities that seasonal fluctuations may cause. Among these, Namba and Takahashi (13) review the results on Lotka-Volterra competition systems with periodic coefficients, and show the new modes of the possibilities of stable periodic solutions even when the stable coexistence cannot be realized in the corresponding classical Lotka-Volterra system with constant coefficients. Here, we restrict the competition parameters so that there is no possibility of the multi- ple periodic solutions that (13) shows. Then we consider the situation where the interacting populations experience pseudo-stochastic environmental fluctuations with unpredictable dis- continuous change, such as seasonality in a year with 'a cycle of three cold days and four warm days'. In a separate paper (6), we analyze the Lotka-Volterra competition system with constant coefficients under telegraph noises, i.e., environmental variability causes parameter switching between two systems. Our focus of attention is on the intermediate case where environments have both pre- dictable and unpredictable aspects. This case is studied by using a combined system of two Lotka-Volterra systems with periodic coefficients. In this system, it is assumed that at ev- ery moment the population dynamics is governed by one of the two Lotka-Volterra systems with periodic coefficients. That is, the populations usually experience predictable changes of environments. However, it is also assumed that the population dynamics abruptly becomes governed by another Lotka-Volterra system. This abrupt switch between two systems occurs
TL;DR: In this article, a sufficient condition for a collection of two-point sets to have the uniqueness property for meromorphic functions has been given, and the condition is shown to hold for all sets.
Abstract: We give a sufficient condition for a collection of two-point sets to have the uniqueness property for meromorphic functions.
TL;DR: In this paper, the authors derive variational formulas of natural first order functionals and obtain criteria for stability at Riemannian subimmersions, in particular at submersions of first order functions.
Abstract: We derive variational formulas of natural first order functionals and obtain
criteria for stability in particular at Riemannian subimmersions.
TL;DR: In this article, the authors define functions which are monotone close to boundary and some oscillation estimates for these functions in Orlicz classes are given for monotonicity close to the boundary.
Abstract: Functions which are monotone close to boundary are defined. Some oscillation estimates are given for these functions in Orlicz classes. Criteria for monotonicity close to boundary are obtained.
TL;DR: In this article, a class of isolated hypersurface singularities, the so-called purely elliptic singularities of complex algebraic varieties of dimension greater than or equal to two, is investigated.
Abstract: We investigate a class of isolated hypersurface singularities, the so-called purely elliptic singularities, of complex algebraic varieties of dimension greater than or equal to two. We show that, for hypersurface purely elliptic singularities defined by nondegenerate polynomials, Calabi-Yau varieties arising among the irreducible components of the essential divisors are concretely associated with the defining equations of these singularities, and that the birational class of the Calabi-Yau varieties does not depend on the irreducible components.
TL;DR: In this article, the Dirichlet problem and the initial-boundary value problem of elliptic and parabolic equations with two power nonlinearities were considered and the existence and nonexistence of positive solutions to these problems on bounded and unbounded domains were obtained by using the results of quenching problem and sub-super solution method.
Abstract: We consider, respectively, the Dirichlet problem and the initial-boundary value problem of elliptic and parabolic equations with two power nonlinearities. We find that these problems are closely related to the so-called quenching problem. We obtain the existence and nonexistence of positive solutions to these problems on bounded and unbounded domains, by using the results of quenching problem and sub-super solution method.