Showing papers in "Tohoku Mathematical Journal in 2009"
TL;DR: In this article, it was shown that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field.
Abstract: We prove that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field. Moreover, we classify a real hypersurface admitting so-called “$\eta$-Ricci soliton” in a non-flat complex space form.
154 citations
TL;DR: In this paper, the authors completely classified parallel non-degenerate surfaces in 4-dimensional Lorentzian space forms, and they also classified non-decentralized surfaces with parallel mean curvature vector.
Abstract: In this paper we completely classify parallel non-degenerate surfaces in 4-dimensional Lorentzian space forms. In addition, we also completely classify non-degenerate surfaces with parallel mean curvature vector in 4-dimensional Lorentzian space forms.
47 citations
TL;DR: In this article, the effective base point free theorem for kawamata log terminal pairs is generalized to log canonical pairs, which was used in Hacon-McKernan's proof of pl flips.
Abstract: Kollar's effective base point free theorem for kawamata log terminal pairs is very important and was used in Hacon-McKernan's proof of pl flips. In this paper, we generalize Kollar's theorem for log canonical pairs.
31 citations
TL;DR: In this article, the degeneracy of integral points on affine 3-fold affine curves with seven ample divisors at infinity was shown to be equivalent to Schmidt's subspace theorem in dimension three or higher.
Abstract: We prove sufficient conditions for the degeneracy of integral points on certain threefolds and other varieties of higher dimension. In particular, under a normal crossings assumption, we prove the degeneracy of integral points on an affine threefold with seven ample divisors at infinity. Analogous results are given for holomorphic curves. As in our previous works [2], [5], the main tool involved is Schmidt's Subspace Theorem, but here we introduce a technical novelty which leads to stronger results in dimension three or higher.
30 citations
TL;DR: In this paper, a set-valued stochastic differential equation with a drift term and a single valued diffusion term is considered in the M-type 2-Banach space and strong solutions which are continuous in the Hausdorff distance are proved.
Abstract: In a certain Banach space called an M-type 2 Banach space (including Hilbert spaces), we consider a set-valued stochastic differential equation with a set-valued drift term and a single valued diffusion term, under the Lipschitz continuity conditions, and we prove the existence and uniqueness of strong solutions which are continuous in the Hausdorff distance.
27 citations
TL;DR: In this paper, the continuity properties of Riesz potentials are studied in the context of Sobolev type inequalities for functions in Orlicz classes, where the continuity property is investigated.
Abstract: In this paper we are concerned with Sobolev type inequalities for Riesz potentials of functions in Orlicz classes. As an application, we study continuity properties of Riesz potentials.
23 citations
TL;DR: In this paper, the authors use primitive collections to characterize the cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case.
Abstract: This paper studies Batyrev’s notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with an open problem.
17 citations
TL;DR: In this article, a model of branching Brownian motions in random environment associated with the Poisson random measure is considered and a relation between the slow population growth and the localization property in terms of the replica overlap is found.
Abstract: We consider a model of branching Brownian motions in random environment associated with the Poisson random measure. We find a relation between the slow population growth and the localization property in terms of the replica overlap. Applying this result, we prove that, if the randomness of the environment is strong enough, this model possesses the strong localization property, that is, particles gather together at small sets.
15 citations
TL;DR: In this article, a generalization of Cheeger-Colding's suspension theorem for manifolds with almost maximal diameters is given, and a relationship between the eigenvalues of the Laplacian and the structure of tangent cones of noncollapsing limit spaces is discussed.
Abstract: In this paper, we give a generalization of Cheeger-Colding's suspension theorem for manifolds with almost maximal diameters. We also discuss a relationship between the eigenvalues of the Laplacian and the structure of tangent cones of non-collapsing limit spaces.
15 citations
TL;DR: In this paper, the authors considered a complete nonsingular complex algebraic variety having a normal crossing divisor such that the associated logarithmic tangent bundle is generated by its global sections.
Abstract: We consider a complete nonsingular complex algebraic variety having a normal crossing divisor such that the associated logarithmic tangent bundle is generated by its global sections. We obtain an optimal vanishing theorem for logarithmic Dolbeault cohomology of nef line bundles in that setting. This implies a vanishing theorem for ordinary Dolbeault cohomology which generalizes results of Broer for flag varieties, and of Mavlyutov for toric varieties.
13 citations
TL;DR: In this paper, the analytic subvariety of the second cohomology group of compact complex Fano manifolds with reductive Lie algebras of holomorphic vector fields was determined.
Abstract: For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of Kahler classes whose Bando-Calabi-Futaki character vanishes. Then a Kahler class contains a Kahler metric of constant scalar curvature if and only if the Kahler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic Kahler classes containing Kahler metrics of constant scalar curvature but does not admit any Kahler-Einstein metric.
TL;DR: In this article, it was shown that there are non-full harmonic maps from the 2-sphere to the 3-space and 4-space which have non-integrable Jacobi fields.
Abstract: In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to asmooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sphere which have non-integrable Jacobi fields. This is particularly surprising in the case of the 3-sphere where the space of harmonic maps of any degree is a smooth manifold, each map having image in a totally geodesic 2-sphere.
TL;DR: In this article, the authors considered the case where a locally nilpotent derivation of a log affine pseudoplane is a locally invariant subalgebra of a normal affine domain.
Abstract: Let $B$ be a normal affine $\boldsymbol{C}$-domain and let $A$ be a $\boldsymbol{C}$-subalgebra of $B$ such that $B$ is a finite $A$-module. Let $\delta$ be a locally nilpotent derivation on $A$. Then $\delta$ lifts uniquely to the quotient field $L$ of $B$, which we denote by $\Delta$. We consider when $\Delta$ is a locally nilpotent derivation of $B$. This is a classical subject treated in [17, 19, 16]. We are interested in the case where $A$ is the $G$-invariant subring of $B$ when a finite group $G$ acts on $B$. As a related topic, we treat in the last section the finite coverings of log affine pseudo-planes in terms of the liftings of the $\boldsymbol{A}^1$-fibrations associated with locally nilpotent derivations.
TL;DR: An infinitesimal derivative of the Bott class is defined by generalizing Heitsch'es construction in this article, which gives a generalization of the Maszczyk formula for the Godbillon-Vey class of real codimension-one foliations.
Abstract: An infinitesimal derivative of the Bott class is defined by generalizing Heitsch'es construction. We prove a formula relating the infinitesimal derivative to the Schwarzian derivatives, which gives a generalization of the Maszczyk formula for the Godbillon-Vey class of real codimension-one foliations. As an application, a residue of infinitesimal derivatives with respect to the Julia set in the sense of Ghys, Gomez-Mont and Saludes is introduced.
TL;DR: In this article, the authors analyze the canonical mixed Hodge structure on the cohomology of these quasi-projective varieties and determine the successive quotients of the corresponding weight filtration.
Abstract: We study arithmetic varieties $V$ attached to certain inner forms of $\boldsymbol{Q}$-rank one of the split symplectic $\boldsymbol{Q}$-group of degree two. These naturally arise as unitary groups of a 2-dimensional non-degenerate Hermitian space over an indefinite rational quaternion division algebra. First, we analyze the canonical mixed Hodge structure on the cohomology of these quasi-projective varieties and determine the successive quotients of the corresponding weight filtration. Second, by interpreting the cohomology groups within the framework of the theory of automorphic forms, we determine the internal structure of the cohomology “at infinity” of $V$, that is, the part which is spanned by regular values of suitable Eisenstein series or residues of such. In conclusion, we discuss some relations between the mixed Hodge structure and the so called Eisenstein cohomology. For example, we show that the Eisenstein cohomology in degree two consists of algebraic cycles.
TL;DR: In this article, the authors studied large deviation principle for random upper semicontinuous functions and obtained Cramer type theorems for those whose underlying space is a separable Banach space of type $p$.
Abstract: In this paper, we shall study large deviation principle for random upper semicontinuous functions, and obtain Cramer type theorems for those whose underlying space is a separable Banach space of type $p$.
TL;DR: In this paper, it was shown that the outer automorphism group of a lattice of a Lie group is represented by a real algebraic number field, up to commensurability.
Abstract: We consider solvable Lie groups which are isomorphic to unimodularizations of products of affine groups. It is shown that a lattice of such a Lie group is determined, up to commensurability, by a totally real algebraic number field. We also show that the outer automorphism group of the lattice is represented faithfully in the automorphism group of the number field. As an application, we obtain a classification of codimension one, volume preserving, locally free actions of products of affine groups.
TL;DR: In this article, the authors studied the local triviality of the ideal class group of the rational field extension over any prime number, and obtained a known general result in such study by modifying the proof of the result.
Abstract: For any prime number $p$, we study local triviality of the ideal class group of the ${\boldsymbol Z}_p$-extension over the rational field. We improve a known general result in such study by modifying the proof of the result, and pursue known effective arguments on the above triviality with the help of a computer. Some explicit consequences of our investigations are then provided in the case $p\leq7$.
TL;DR: In this article, the relation between pseudo-Hermitian symmetric pairs and para-hermitian ones was investigated and a relation between the two classes of symmetric pair was established.
Abstract: In this paper, we investigate relation between pseudo-Hermitian symmetric pairs and para-Hermitian symmetric ones.
TL;DR: In this article, the authors proved the boundedness of the singular integrals with rough variable kernels for a class of modified directional Hilbert transform (DHT) with rough odd kernels.
Abstract: We prove the $L^p (1
TL;DR: In this paper, the authors defined tangent lightcone map, tangentlightcone curvature and height function and studied the geometry of the timelike surfaces in Minkowski 4-space through their contact with spacelike hyperplane.
Abstract: In the paper, we will define tangent lightcone map, tangentlightcone curvature and tangent lightcone height function. Then we study the geometry of the timelike surfaces in Minkowski 4-space through their contact with spacelike hyperplane and give the classification of singularities of tangent lightcone map based on the Legendrian singularity theory of Arnol'd.
TL;DR: In this paper, it was shown that the Laplacian acting on differential forms on a sphere can be lifted to an operator on its rotation group, which is intrinsically equivalent to the LAP acting on functions on the Lie group.
Abstract: We show that the Laplacian acting on differential forms on a sphere can be lifted to an operator on its rotation group which is intrinsically equivalent to the Laplacian acting on functions on the Lie group. Further, using the result and the Urakawa summation formula for the heat kernel of the latter Laplacian and the Weyl integration formula, we get a summation formula for the kernel of the former.
TL;DR: In this article, the authors considered a nonlinear elliptic equation on the plane away from the origin, which arises from the spherical Onsager vortex theory in physics or the prob- lem of prescribing Gaussian curvature in geometry.
Abstract: In this paper, we consider a nonlinear elliptic equation on the plane away from the origin, which arises from the spherical Onsager vortex theory in physics or the prob- lem of prescribing Gaussian curvature in geometry. Depending on various situations for the prescribed function in the nonlinear term, the complete structure of radial solutions in terms of initial data will be offered.
TL;DR: In this article, a pair of real-valued functions satisfying a system of second order partial differential equations of hyperbolic type is assosiated with a realvalued function satisfying the vanishing normal curvature.
Abstract: Every isometric immersion of ${\boldsymbol R}^2$ into ${\boldsymbol R}^4$ with vanishing normal curvature is assosiated with a pair of real-valued functions satisfying a system of second order partial differential equations of hyperbolic type,and vice versa. An isometric immersion with vanishing normal curvature is revealed to be multiple-valued in general as is shown by some concrete examples.
TL;DR: In this paper, a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the Yamabe operator was given for a transverse spin foliation.
Abstract: On a transverse spin foliation, we give a new lower bound for the square of the eigenvalues of the basic Dirac operator by the smallest eigenvalue of the basic Yamabe operator. Moreover, the limiting foliation is transversally Einsteinian.
TL;DR: In this article, the inner product of an Eisenstein series on a unitary group and a nonholomorphic cuspidal Hecke eigenform restricted to a subgroup was computed to obtain an integral representation of the standard $L$-function.
Abstract: We consider the real rank one unitary group $G$ and its subgroup $H$ obtained as the stabilizer of an anisotropic vector in the skew-hermitian space defining $G$. We compute the inner-product of an Eisenstein series on $H$ and a non-holomorphic cuspidal Hecke eigenform on $G$ restricted to $H$ to obtain an integral representation of the standard $L$-function of the eigenform. We also discuss some consequences of the integral representation.