Showing papers in "Tohoku Mathematical Journal in 2012"
TL;DR: In this article, a new representation of the Schrodinger operator of a free particle by using the short-time Fourier transform was proposed, and its applications were described. But this representation is restricted to the case of a single particle.
Abstract: We propose a new representation of the Schrodinger operator of a free particle by using the short-time Fourier transform and give its applications.
38 citations
TL;DR: In this article, the existence of minimizers of the Hardy-Sobolev type variational problem was studied and it was shown that the Hardy Sobolev best constant admits its minimizers provided the bounded smooth domain has the negative mean curvature at the origin on the boundary.
Abstract: In this paper, we consider the existence of minimizers of the Hardy-Sobolev type variational problem. Recently, Ghoussoub and Robert (9, 10) proved that the Hardy- Sobolev best constant admits its minimizers provided the bounded smooth domain has the negative mean curvature at the origin on the boundary. We generalize their results by using the idea of Brezis and Nirenberg, and as a consequence, we shall prove the existence of pos- itive solutions to the elliptic equation involving two different kinds of Hardy-Sobolev critical exponents.
30 citations
TL;DR: In this article, the authors investigate singularities of all parallel surfaces to a given regular surface and give criteria for these singularity types in terms of differential geometry (Theorems 3.4 and 3.5).
Abstract: We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional $D_4^\pm$ singularities. We give criteria for these singularity types in terms of differential geometry (Theorems 3.4 and 3.5).
28 citations
TL;DR: In this article, it was shown that a topological building that is compact and totally disconnected can be given the structure of a locally finite affine building when Delta is the building at infinity of such a building.
Abstract: Let Delta be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that Delta can be given the structure of a topological building that is compact and totally disconnected precisely when Delta is the building at infinity of a locally finite affine building.
18 citations
TL;DR: In this paper, the authors study Tauberian conditions for the existence of Cesaro limits in terms of the Laplace transform and derive new Tauberians for the converse of Abel's theorem where the conclusion is Cesaro summability rather than convergence.
Abstract: We study Tauberian conditions for the existence of Cesaro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber's second theorem on the converse of Abel's theorem. For Schwartz distributions, we obtain extensions of many classical Tauberians for Cesaro and Abel summability of functions and measures. We give general Tauberian conditions in order to guarantee $(\mathrm{C},\beta)$ summability for a given order $\beta$. The results are directly applicable to series and Stieltjes integrals, and we therefore recover the classical cases and provide new Tauberians for the converse of Abel's theorem where the conclusion is Cesaro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Szasz for Dirichlet series.
13 citations
TL;DR: In this paper, the characterization of maximum dimensional proper-bihar-monic integral C-parallel submanifolds of a Sasakian space form and a 7-dimensional SasakIAN space form is presented.
Abstract: We find the characterization of maximum dimensional proper-bihar- monic integral C-parallel submanifolds of a Sasakian space form and then classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere S 7 we explicitly find all 3-dimensional proper-biharmonic integral C-parallel sub- manifolds. We also determine the proper-biharmonic parallel Lagrangian sub- manifolds of CP 3 .
13 citations
TL;DR: In this paper, the authors classify cohomogeneity of one special Lagrangian submanifolds in the cotangent bundle of the sphere $S n$ invariant under the Stenzel metric and a Ricci-flat cone Kahler metric.
Abstract: We classify cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere $S^n$ invariant under $SO(p) \times SO(n+1-p)$ with respect to the Stenzel metric and a Ricci-flat cone Kahler metric. Moreover, we describe the asymptotic behavior and singularities of such special Lagrangian submanifolds.
10 citations
TL;DR: In this paper, the universal cover of a typical leaf is defined in such a way that the measure $m$ is described in terms of these functions $h$ on various leaves.
Abstract: Given a harmonic measure $m$ of a hyperbolic lamination $\mathcal L$ on a compact metric space $M$, a positive harmonic function $h$ on the
universal cover of a typical leaf is defined in such a way that the measure $m$ is described in terms of these functions $h$ on various leaves. We
discuss some properties of the function $h$. We show that if $m$ is ergodic and not completely invariant, then $h$ is typically unbounded and is
induced by a probability $\mu$ of the sphere at infinity which is singular to the Lebesgue measure. A harmonic measure is called Type I (resp. Type II)
if for any typical leaf, the measure $\mu$ is a point mass (resp. of full support). We show that any ergodic harmonic measure is either of type I or type II.
9 citations
TL;DR: In this article, the authors classify explicitly pairs $(S, G)$ with automorphisms of order four of a complex nonsingular minimal projective surface of general type with q(S) = 2.
Abstract: Let $S$ be a complex nonsingular minimal projective surface of general type with $q(S)=2$, and let $G$ be the group of the automorphisms of $S$ acting trivially on $H^2(S, \boldsybmol{Q})$. In this note we classify explicitly pairs $(S, G)$ with $G$ of order four.
9 citations
8 citations
TL;DR: In this paper, it was shown that every reflective submanifold of a symmetric $R$-space is convex, and that the same is true for all reflective submansifolds.
Abstract: We show that every reflective submanifold of a symmetric $R$-space is (geodesically) convex.
TL;DR: In this article, the authors parametrize the space of isometric immersions of the hyper-bolic plane into the hyperbolic 3-space in terms of null-causal curves.
Abstract: In this paper, we parametrize the space of isometric immersions of the hyper- bolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of ori- ented geodesics. Moreover, we characterize "ideal cones" (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.
TL;DR: In this paper, the authors investigated the rate of the approximation by Cesaro means of the quadratical partial sums of double Walsh-Kaczmarz-Fourier series of a function in the Lebesgue space over the Walsh group.
Abstract: In this article we investigate the rate of the approximation by Cesaro means of the quadratical partial sums of double Walsh-Kaczmarz-Fourier series of a function in the Lebesgue space over the Walsh group. The approximation properties of Cesaro means of negative order of one- and two-dimensional Walsh-Fourier series was discussed earlier by Goginava.
TL;DR: In this paper, the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure was proved.
Abstract: In this paper, we shall prove the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure. This generalizes the result of Cho, Futaki and Ono for compact toric Sasaki manifolds.
TL;DR: The conjecture of Blair as mentioned in this paper that there are no nonflat Riemannian metrics of nonpositive curvature associated with a contact structure has been proved for a certain class of contact structures on closed 3-dimensional manifolds.
Abstract: The conjecture of Blair says that there are no nonflat Riemannian metrics of nonpositive curvature associated with a contact structure. We prove this conjecture for a certain class of contact structures on closed 3-dimensional manifolds and construct a local counterexample.
TL;DR: In this paper, the geometric invariant theory is developed for actions of linearly reductive group schemes on formal affine schemes, and conditions for when the existence of good moduli spaces can be deduced from the presence of etale charts.
Abstract: We study the local properties of Artin stacks and their good moduli spaces, if they exist. We show that near closed points with linearly reductive stabilizer, Artin stacks formally locally admit good moduli spaces. In particular, the geometric invariant theory is developed for actions of linearly reductive group schemes on formal affine schemes. We also give conditions for when the existence of good moduli spaces can be deduced from the existence of etale charts admitting good moduli spaces.
TL;DR: In this article, a technique for constructing contact structures compatible with fibered Seifert multilinks in homology 3-spheres and fibered multilink with cabling structures in any 3-manifolds is introduced.
Abstract: We introduce a new technique for constructing contact structures compatible with fibered Seifert multilinks in homology 3-spheres and fibered multilinks with cabling structures in any 3-manifolds. The contact structure constructed in this paper is so well-organized that we can easily see if it is tight or overtwisted in most cases. As an application, we classify the tightness of contact structures compatible with fibered Seifert links in S. We also give an application to the fibrations of real analytic germs of the form fḡ.
TL;DR: In this paper, a generalization of the results concerning liftings and Fourier coefficients of Jacobi forms to the case of elliptic modular forms of index N$ over totally real number fields is presented.
Abstract: Skoruppa and Zagier established a bijective correspondence from the space of Jacobi forms $\phi$ of index $m$ to that of elliptic modular forms $f$ of level $m$. Gross, Kohnen and Zagier formulated this correspondence by means of kernel functions. Moreover, they proved that the squares of Fourier coefficients of $\phi$ are essentially equal to the critical values of the zeta functions $L(s,f,\chi)$ of $f$ twisted by a quadratic character $\chi$. The purpose of this paper is to prove a generalization of such results concerning liftings and Fourier coefficients of Jacobi forms to the case of Jacobi forms of index $N$ over totally real number fields $F$. Using kernel functions associated with the space of quadratic forms, we shall establish the existence of a lifting from the space of Jacobi forms $\phi$ of index $N$ over $F$ to that of Hilbert modular forms $f$ of level $N$ over $F$. Moreover, we determine explicitly the Fourier coefficients of $f$ from those of $\phi$. We prove that an analogue of Waldspurger's theorem in the case of Jacobi forms of index $N$ over $F$ holds.
TL;DR: In this article, the authors studied the topologies of the direct limits of the sequence in the categories of topological groups and of uniform spaces and found conditions under which these two direct limit topologies coincide.
Abstract: Given an increasing sequence $(G_n)$ of topological groups, we study the topologies of the direct limits of the sequence $(G_n)$ in the categories of topological groups and of uniform spaces and find conditions under which these two direct limit topologies coincide.
TL;DR: In this paper, the slope invariants for the resulting (1, 1)-knots were derived for genus-1 1-bridge knots in the 3-sphere.
Abstract: For a genus-1 1-bridge knot in the 3-sphere, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. We generalize their construction and calculate the slope invariants for the resulting middle tunnels. In particular, we obtain the slope sequence of the original example of Goda, Hayashi, and Ishihara.
TL;DR: In this paper, the authors consider a reflexive module of rank one over a degenerate Weyl algebra over a field of positive characteristic and define an invariant which they call wrinkle of the module and see that it is good enough to distinguish trivial module.
Abstract: We consider a reflexive module of rank one over a degenerate Weyl algebra over a field of positive characteristic. We define an invariant which we call wrinkle of the module and see that it is good enough to distinguish trivial module.
TL;DR: In this article, the concept of Sato Grassmannians of locally linearly compact vector spaces (Tate spaces) was generalized to the Beilinson category of generalized Tate spaces.
Abstract: We generalize the concept of Sato Grassmannians of locally linearly compact topological vector spaces (Tate spaces) to the Beilinson category of
the “locally compact objects”, or Generalized Tate Spaces, of an exact category. This allows us to extend the Kapranov dimensional
torsor Dim and determinantal gerbe Det to generalized Tate spaces and unify their treatment in the determinantal torsor. We then introduce a class of exact
categories, that we call partially abelian exact, and prove that if the base category is so, then Dim and Det are multiplicative in admissible short exact sequences
of generalized Tate spaces. We then give a cohomological interpretation of these results in terms of the Waldhausen K-theoretical space of the Beilinson category.
Our approach can be iterated and we define analogous concepts for the successive categories of $n$-dimensional (generalized) Tate spaces. In particular we show
that the category of Tate spaces is partially abelian exact, so we can extend the results for Dim and Det obtained for 1-Tate spaces to 2-Tate spaces, and provide
a new interpretation in the context of algebraic $K$-theory of results of Kapranov, Arkhipov-Kremnizer and Frenkel-Zhu.
TL;DR: In this paper, the authors studied the property of Nevanlinna's deficiency as function on linear systems in smooth complex projective algebraic varieties and gave a structure theorem for the set of deficient divisors.
Abstract: We study property of Nevanlinna's deficiency as functions on linear systems in smooth complex projective algebraic varieties. We give a structure theorem for the set of deficient divisors. This structure theorem yields that the set of values of deficiency is at most countable. Moreover, we have a correspondence between the deficiencies and the linear systems.
TL;DR: In this paper, two generalizations of the Clifford theorem to algebraic surfaces are presented, and some bounds for the number of moduli of surfaces of general type are obtained for general types.
Abstract: We give two generalizations of the Clifford theorem to algebraic surfaces. As an application, we obtain some bounds for the number of moduli of surfaces of general type.
TL;DR: In this paper, the heat kernel transform on a nilmanifold M associated to a H-type group was studied and a reduction technique was used to reduce the problem to the case of Heisenberg groups.
Abstract: We study the heat kernel transform on a nilmanifold M associated to a H-type group. Using a reduction technique we reduce the problem to the case of Heisenberg groups. The image of $ L^2(M) $ under the heat kernel transform is shown to be a direct sum of weighted Bergman spaces.