Tohoku Mathematical Journal Articles (Project Euclid)

$\boldsymbol{Q}$-factorial Gorenstein toric Fano varieties with large Picard number

Thu, 05 Aug 2010 15:41 EDT

Benjamin Nill, Mikkel Øbro

Source: Tohoku Math. J. (2), Volume 62, Number 1, 1--15.

Abstract:
In dimension $d$, ${\boldsymbol Q}$-factorial Gorenstein toric Fano varieties with Picard number $\rho_X$ correspond to simplicial reflexive polytopes with $\rho_X + d$ vertices. Casagrande showed that any $d$-dimensional simplicial reflexive polytope has at most $3 d$ and $3d-1$ vertices if $d$ is even and odd, respectively. Moreover, for $d$ even there is up to unimodular equivalence only one such polytope with $3 d$ vertices, corresponding to the product of $d/2$ copies of a del Pezzo surface of degree six. In this paper we completely classify all $d$-dimensional simplicial reflexive polytopes having $3d-1$ vertices, corresponding to $d$-dimensional ${\boldsymbol Q}$-factorial Gorenstein toric Fano varieties with Picard number $2d-1$. For $d$ even, there exist three such varieties, with two being singular, while for $d > 1$ odd there exist precisely two, both being nonsingular toric fiber bundles over the projective line. This generalizes recent work of the second author.

On low pass filters in a frame multiresolution analysis

Tue, 11 Oct 2011 09:16 EDT

Angel San Antolín

Source: Tohoku Math. J. (2), Volume 63, Number 3, 427--439.

Abstract:
We give necessary and sufficient conditions for a measurable function to be a low pass filter associated to a scaling function in a frame multiresolution analysis. Those conditions involve the class of real-valued bounded measurable functions such that the origin is a point of approximate continuity of such functions. The main result here is proved in a general context where the considered dilation is given by a fixed expansive linear map.

On a class of foliated non-Kählerian compact complex surfaces

Tue, 11 Oct 2011 09:16 EDT

Marco Brunella

Source: Tohoku Math. J. (2), Volume 63, Number 3, 441--460.

Abstract:
Motivated by recent results on non-Kählerian compact complex surfaces with small second Betti number, we classify those on which a holomorphic foliation (with singularities) exists.

The first hundred years of the Tohoku Mathematical Journal

Fri, 06 Jan 2012 16:45 EST

Tadao Oda

Source: Tohoku Math. J. (2), Volume 63, Number 4, 461--470.

Collapsing three-manifolds with a lower curvature bound

Fri, 06 Jan 2012 16:45 EST

Takashi Shioya

Source: Tohoku Math. J. (2), Volume 63, Number 4, 471--487.

Abstract:
We survey works on collapsing Riemannian manifolds with a lower bound of sectional curvature, focusing on the three-dimensional case. We also explain the basics of Seifert manifolds and Alexandrov spaces quickly and a key idea of our proof of the volume collapsing theorem.

Mass problems associated with effectively closed sets

Fri, 06 Jan 2012 16:45 EST

Stephen G. Simpson

Source: Tohoku Math. J. (2), Volume 63, Number 4, 489--517.

Abstract:
The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov's non-rigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let $\mathcal{E}_\mathrm{w}$ be this lattice. We show that $\mathcal{E}_\mathrm{w}$ provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in $\mathcal{E}_\mathrm{w}$ which are associated with such problems. In addition, we present some structural results concerning the lattice $\mathcal{E}_\mathrm{w}$. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how $\mathcal{E}_\mathrm{w}$ can be applied in symbolic dynamics, toward the classification of tiling problems and $\boldsymbol{Z}^d$-subshifts of finite type.

Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions

Fri, 06 Jan 2012 16:45 EST

Christopher D. Sogge

Source: Tohoku Math. J. (2), Volume 63, Number 4, 519--538.

Abstract:
We provide a necessary and sufficient condition that $L^p$-norms, $2<p<$, of eigenfunctions of the square root of minus the Laplacian on two-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $\lambda$. The condition that ensures this is that their $L^2$-norms over $O(\lambda^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $\lambda$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). The proof exploits Gauss' lemma and the fact that the bilinear oscillatory integrals in Hörmander's proof of the Carleson-Sjölin theorem become better and better behaved away from the diagonal. Our results are related to a recent work of Bourgain who showed that $L^2$-averages over geodesics of eigenfunctions are small compared to a natural power of the eigenvalue $\lambda$ provided that the $L^4(M)$ norms are similarly small. Our results imply that QUE cannot hold on a compact boundaryless Riemannian manifold $(M,g)$ of dimension two if $L^p$-norms are saturated for a given $2<p<6$. We also show that eigenfunctions cannot have a maximal rate of $L^2$-mass concentrating along unit portions of geodesics that are not smoothly closed.

$K$-finite solutions to conformally invariant systems of differential equations

Fri, 06 Jan 2012 16:45 EST

Anthony C. Kable

Source: Tohoku Math. J. (2), Volume 63, Number 4, 539--559.

Abstract:
Let $G$ be a connected semisimple linear real Lie group, and $Q$ (resp. $K$) a real parabolic subgroup (resp. maximal compact subgroup) of $G$. The space of $K$-finite solutions to a conformally invariant system of differential equations on a line bundle over the real flag manifold $G/Q$ is studied. The general theory is then applied to certain second order systems on the flag manifold that corresponds to the Heisenberg parabolic subgroup in a split simple Lie group.

Homoclinic and heteroclinic orbits for a semilinear parabolic equation

Fri, 06 Jan 2012 16:45 EST

Marek Fila, Eiji Yanagida

Source: Tohoku Math. J. (2), Volume 63, Number 4, 561--579.

Abstract:
We study the existence of connecting orbits for the Fujita equation with a critical or supercritical exponent. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and a homoclinic orbit with respect to zero.

The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces

Fri, 06 Jan 2012 16:45 EST

Victor Batyrev, Mark Blume

Source: Tohoku Math. J. (2), Volume 63, Number 4, 581--604.

Abstract:
A root system $R$ of rank $n$ defines an $n$-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system $R$ and apply this result in the case of root systems of type $A$ to give a new proof of the fact that the toric variety $X(A_n)$ is the fine moduli space $\overline{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.

Ricci curvature of graphs

Fri, 06 Jan 2012 16:45 EST

Yong Lin, Linyuan Lu, Shing-Tung Yau

Source: Tohoku Math. J. (2), Volume 63, Number 4, 605--627.

Abstract:
We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

On nef and semistable hermitian lattices, and their behaviour under tensor product

Fri, 06 Jan 2012 16:45 EST

Yves André

Source: Tohoku Math. J. (2), Volume 63, Number 4, 629--649.

Abstract:
We study the behaviour of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces. One approach to show that semistable vector bundles in characteristic zero are preserved by tensor product is based on the notion of nef vector bundles. We revisit this approach and show how far it can be transferred to hermitian lattices. J.-B. Bost conjectured that semistable hermitian lattices are preserved by tensor product. Using properties of nef hermitian lattices, we establish an inequality in that direction. We axiomatize our method in the general context of monoidal categories, and then give an elementary proof of the fact that semistable multifiltered vector spaces (which play a role in diophantine approximation) are preserved by tensor product.

Stability of a stationary solution for the Lugiato-Lefever equation

Fri, 06 Jan 2012 16:45 EST

Tomoyuki Miyaji, Isamu Ohnishi, Yoshi Tsutsumi

Source: Tohoku Math. J. (2), Volume 63, Number 4, 651--663.

Abstract:
We study the stability of a stationary solution for the Lugiato-Lefever equation with the periodic boundary condition in one space dimension, which is a damped and driven nonlinear Schrödinger equation introduced to model the optical cavity. In this paper, we prove the Strichartz estimates for the linear damped Schrödinger equation with potential and external forcing and investigate the stability of certain stationary solutions under the initial perturbation within the framework of $L^2$.

Web Markov skeleton processes and their applications

Fri, 06 Jan 2012 16:45 EST

Yuting Liu, Zhi-Ming Ma, Chuan Zhou

Source: Tohoku Math. J. (2), Volume 63, Number 4, 665--695.

Abstract:
We propose and discuss a new class of processes, web Markov skeleton processes (WMSP), arising from the information retrieval on the Web. The framework of WMSP covers various known classes of processes, and it contains also important new classes of processes. We explore the definition, the scope and the time homogeneity of WMSPs, and discuss in detail a new class of processes, mirror semi-Markov processes. In the last section we briefly review some applications of WMSPs in computing page importance on the Web.

Counting pseudo-holomorphic discs in Calabi-Yau 3-holds

Fri, 06 Jan 2012 16:45 EST

Kenji Fukaya

Source: Tohoku Math. J. (2), Volume 63, Number 4, 697--727.

Abstract:
In this paper we define an invariant of a pair of a 6 dimensional symplectic manifold with vanishing 1st Chern class and its relatively spin Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of bounding cochains (solutions of the $A_{\infty}$ version of the Maurer-Cartan equation of the filtered $A_{\infty}$ algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant. This invariant depends on the choice of almost complex structures. The way how it depends on the almost complex structures is described by a wall crossing formula which involves a moduli space of pseudo-holomorphic spheres.

Quasi-free actions of finite groups on the Cuntz algebra $\mathcal{O}_\infty$

Fri, 06 Jan 2012 16:45 EST

Pavle Goldstein, Masaki Izumi

Source: Tohoku Math. J. (2), Volume 63, Number 4, 729--749.

Abstract:
We show that any faithful quasi-free actions of a finite group on the Cuntz algebra $\mathcal{O}_\infty$ are mutually conjugate, and that they are asymptotically representable.

Blow-ups and mixed motives

Fri, 06 Jan 2012 16:45 EST

Masaki Hanamura

Source: Tohoku Math. J. (2), Volume 63, Number 4, 751--774.

Abstract:
The blow-up formula for Chow groups of smooth varieties is known; for smooth projective varieties there is a similar formula for motives. We generalize these and prove blow-up formulas for higher Chow groups and for mixed motives of smooth quasi-projective varieties.

Ramification and cleanliness

Fri, 06 Jan 2012 16:45 EST

Ahmed Abbes, Takeshi Saito

Source: Tohoku Math. J. (2), Volume 63, Number 4, 775--853.

Abstract:
This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell \ne p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth, separated and quasi-compact $k$-scheme, $D$ a simple normal crossing divisor on $X$, $U=X-D$, $\Lambda$ a finite local ${\mathbb Z}_{\ell} $-algebra and ${\mathscr F}$ a locally constant constructible sheaf of $\Lambda$-modules on $U$. We introduce a {\em boundedness} condition on the ramification of ${\mathscr F}$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the {\em characteristic cycle} of ${\mathscr F}$. The cleanliness condition extends the one introduced by Kato for rank 1 sheaves. Roughly speaking, it means that the ramification of ${\mathscr F}$ along $D$ is controlled by its ramification at the generic points of $D$. Under this condition, we propose a conjectural Riemann-Roch type formula for ${\mathscr F}$. Some cases of this formula have been previously proved by Kato and by the second author (T. S.).

Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds

Fri, 06 Jan 2012 16:45 EST

Alessio Figalli, Ludovic Rifford, Cédric Villani

Source: Tohoku Math. J. (2), Volume 63, Number 4, 855--876.

Abstract:
In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma-Trudinger-Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity, and review existing results.

Small noise asymptotic expansions for stochastic PDE's, I. The case of a dissipative polynomially bounded non linearity

Fri, 06 Jan 2012 16:45 EST

Sergio Albeverio, Luca Di Persio, Elisa Mastrogiacomo

Source: Tohoku Math. J. (2), Volume 63, Number 4, 877--898.

Abstract:
We study a reaction-diffusion evolution equation perturbed by a Gaussian noise. Here the leading operator is the infinitesimal generator of a $C_0$-semigroup of strictly negative type, the nonlinear term has at most polynomial growth and is such that the whole system is dissipative. The corresponding Itô stochastic equation describes a process on a Hilbert space with dissipative nonlinear, non globally Lipschitz drift and a Gaussian noise. Under smoothness assumptions on the nonlinearity, asymptotics to all orders in a small parameter in front of the noise are given, with uniform estimates on the remainders. Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular example we consider the small noise asymptotic expansions for the stochastic FitzHugh-Nagumo equations of neurobiology around deterministic solutions.

Direct limit topologies in the categories of topological groups and of uniform spaces

Mon, 26 Mar 2012 09:09 EDT

Taras Banakh, Dušan Repovš

Source: Tohoku Math. J. (2), Volume 64, Number 1, 1--24.

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.

Compatible contact structures of fibered Seifert links in homology 3-spheres

Mon, 26 Mar 2012 09:09 EDT

Masaharu Ishikawa

Source: Tohoku Math. J. (2), Volume 64, Number 1, 25--59.

Abstract:
We study compatible contact structures of fibered Seifert multilinks in homology 3-spheres and especially give a necessary and sufficient condition for the contact structure to be tight in the case where the Seifert fibration is positively twisted. As a corollary we determine the strongly quasipositivity of fibered Seifert links in $S^3$. We also study the compatible contact structures of cablings along links in any 3-manifolds.

Norm based extension of reflexive modules over Weyl algebras

Mon, 26 Mar 2012 09:09 EDT

Yoshifumi Tsuchimoto

Source: Tohoku Math. J. (2), Volume 64, Number 1, 61--77.

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.

Minimizing problems for the Hardy-Sobolev type inequality with the singularity on the boundary

Mon, 26 Mar 2012 09:09 EDT

Chang-Shou Lin, Hidemitsu Wadade

Source: Tohoku Math. J. (2), Volume 64, Number 1, 79--103.

Abstract:
In this paper, we consider the existence of minimizers of the Hardy-Sobolev type variational problem. Recently, Ghoussoub and Robert 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 Brézis and Nirenberg, and as a consequence, we shall prove the existence of positive solutions to the elliptic equation involving two different kinds of Hardy-Sobolev critical exponents.

Local properties of good moduli spaces

Mon, 26 Mar 2012 09:09 EDT

Jarod Alper

Source: Tohoku Math. J. (2), Volume 64, Number 1, 105--123.

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 étale charts admitting good moduli spaces.

Projective normality of toric 3-folds with non-big adjoint hyperplane sections

Mon, 26 Mar 2012 09:09 EDT

Shoetsu Ogata

Source: Tohoku Math. J. (2), Volume 64, Number 1, 125--140.

Abstract:
Let $L$ be an ample line bundle on a nonsingular toric 3-fold. We show that if the adjoint bundle of $L$ has no global sections, then $L$ is normally generated. Even if the adjoint bundle is effective, it is shown that $L$ is normally generated if it is not big.

Cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere

Mon, 26 Mar 2012 09:09 EDT

Kaname Hashimoto, Takashi Sakai

Source: Tohoku Math. J. (2), Volume 64, Number 1, 141--169.

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 Kähler metric. Moreover, we describe the asymptotic behavior and singularities of such special Lagrangian submanifolds.

Isometric immersions of the hyperbolic plane into the hyperbolic space

Mon, 02 Jul 2012 13:17 EDT

Atsufumi Honda

Source: Tohoku Math. J. (2), Volume 64, Number 2, 171--193.

Abstract:
In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize “ideal cones” (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.

Biharmonic integral $\mathcal{C}$-parallel submanifolds in 7-dimensional Sasakian space forms

Mon, 02 Jul 2012 13:17 EDT

Dorel Fetcu, Cezar Oniciuc

Source: Tohoku Math. J. (2), Volume 64, Number 2, 195--222.

Abstract:
We find the characterization of maximum dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds of a Sasakian space form and then we classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere $\boldsymbol{S}^7$ we explicitly find all 3-dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds. We also determine the proper-biharmonic parallel Lagrangian submanifolds of $\boldsymbol{C}P^3$.

Representation of Schrödinger operator of a free particle via short-time Fourier transform and its applications

Mon, 02 Jul 2012 13:17 EDT

Keiichi Kato, Masaharu Kobayashi, Shingo Ito

Source: Tohoku Math. J. (2), Volume 64, Number 2, 223--231.

Abstract:
We propose a new representation of the Schrödinger operator of a free particle by using the short-time Fourier transform and give its applications.

Square means versus Dirichlet integrals for harmonic functions on Riemann surfaces

Mon, 02 Jul 2012 13:17 EDT

Hiroaki Masaoka, Mitsuru Nakai

Source: Tohoku Math. J. (2), Volume 64, Number 2, 233--259.

Abstract:
We show rather unexpectedly and surprisingly the existence of a hyperbolic Riemann surface $W$ enjoying the following two properties: firstly, the converse of the celebrated Parreau inclusion relation that the harmonic Hardy space $HM_{2}(W)$ with exponent 2 consisting of square mean bounded harmonic functions on $W$ includes the space $HD(W)$ of Dirichlet finite harmonic functions on $W$, and a fortiori $HM_{2}(W)=HD(W)$, is valid; secondly, the linear dimension of $HM_{2}(W)$, hence also that of $HD(W)$, is infinite.

Construction of homogeneous Lagrangian submanifolds in $\boldsymbol{CP}^n$ and Hamiltonian stability

Mon, 02 Jul 2012 13:17 EDT

David Petrecca, Fabio Podestà

Source: Tohoku Math. J. (2), Volume 64, Number 2, 261--268.

Abstract:
We apply the concept of castling transform of prehomogeneous vector spaces to produce new examples of minimal homogeneous Lagrangian submanifolds in the complex projective space. Furthermore we verify the Hamiltonian stability of a low dimensional example that can be obtained in this way.

On the Clifford theorem for surfaces

Mon, 02 Jul 2012 13:17 EDT

Hao Sun

Source: Tohoku Math. J. (2), Volume 64, Number 2, 269--285.

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.

Deficiencies of holomorphic curves in algebraic varieties

Mon, 02 Jul 2012 13:17 EDT

Yoshihiro Aihara

Source: Tohoku Math. J. (2), Volume 64, Number 2, 287--315.

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.

Approximation by Cesàro means of negative order of double Walsh-Kaczmarz-Fourier series

Tue, 11 Sep 2012 09:17 EDT

Károly Nagy

Source: Tohoku Math. J. (2), Volume 64, Number 3, 317--331.

Abstract:
In this article we investigate the rate of the approximation by Cesàro 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 Cesàro means of negative order of one- and two-dimensional Walsh-Fourier series was discussed earlier by Goginava.

Compact totally disconnected Moufang buildings

Tue, 11 Sep 2012 09:17 EDT

Theo Grundhöfer, Linus Kramer, Hendrik Van Maldeghem, Richard M. Weiss

Source: Tohoku Math. J. (2), Volume 64, Number 3, 333--360.

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.

On the Fourier coefficients of Jacobi forms of index $N$ over totally real number fields

Tue, 11 Sep 2012 09:17 EDT

Hisashi Kojima

Source: Tohoku Math. J. (2), Volume 64, Number 3, 361--385.

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.

Singularities of parallel surfaces

Tue, 11 Sep 2012 09:17 EDT

Toshizumi Fukui, Masaru Hasegawa

Source: Tohoku Math. J. (2), Volume 64, Number 3, 387--408.

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).

Some new Zariski pairs of sextic curves

Tue, 11 Sep 2012 09:17 EDT

Bo Wu, Jin-Gen Yang

Source: Tohoku Math. J. (2), Volume 64, Number 3, 409--426.

Abstract:
A topological invariant of reduced sextic curves with simple singularities is given. Twelve new Zariski pairs of sextic curves are determined. Each pair consists of two curves with non-isomorphic fundamental groups.

Algebraic independence results related to pattern sequences in distinct $\langle q,r \rangle$-numeration systems

Tue, 11 Sep 2012 09:17 EDT

Yohei Tachiya

Source: Tohoku Math. J. (2), Volume 64, Number 3, 427--438.

Abstract:
In this paper, we prove the algebraic independence over ${\boldsymbol C}(z)$ of the generating functions of pattern sequences defined in distinct $\langle q,r \rangle$-numeration systems. Our result asserts that any nontrivial linear combination over ${\boldsymbol C}$ of pattern sequences chosen from distinct $\langle q,r \rangle$-numeration systems can not be a linear recurrence sequence. As an application, we give a linear independence over ${\boldsymbol C}$ of the pattern sequences.

Heat kernel transform on nilmanifolds associated to H-type groups

Tue, 11 Sep 2012 09:17 EDT

Aparajita Dasgupta, Sundaram Thangavelu

Source: Tohoku Math. J. (2), Volume 64, Number 3, 439--451.

Abstract:
We study the heat kernel transform on a nilmanifold $ \Gamma \backslash N $ associated to an H-type group. Using a reduction technique we reduce the problem to the case of Heisenberg groups. The image of $ L^2(\Gamma \backslash N) $ under the heat kernel transform is shown to be a direct sum of weighted Bergman spaces.

Uniqueness of Sasaki-Einstein metrics

Tue, 11 Sep 2012 09:17 EDT

Yasufumi Nitta, Ken'ichi Sekiya

Source: Tohoku Math. J. (2), Volume 64, Number 3, 453--468.

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.

Middle tunnels by splitting

Thu, 20 Dec 2012 16:30 EST

Sangbum Cho, Darryl McCullough

Source: Tohoku Math. J. (2), Volume 64, Number 4, 469--488.

Abstract:
For a genus-1 1-bridge knot in $S^3$, 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.

Sato Grassmannians for generalized Tate spaces

Thu, 20 Dec 2012 16:30 EST

Luigi Previdi

Source: Tohoku Math. J. (2), Volume 64, Number 4, 489--538.

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.

On Tauber's second Tauberian theorem

Thu, 20 Dec 2012 16:30 EST

Ricardo Estrada, Jasson Vindas

Source: Tohoku Math. J. (2), Volume 64, Number 4, 539--560.

Abstract:
We study Tauberian conditions for the existence of Cesàro 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 Cesàro 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 Cesàro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Szàsz for Dirichlet series.

A note on the conjecture of Blair in contact Riemannian geometry

Thu, 20 Dec 2012 16:30 EST

Vladimir Krouglov

Source: Tohoku Math. J. (2), Volume 64, Number 4, 561--567.

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.

The dichotomy of harmonic measures of compact hyperbolic laminations

Thu, 20 Dec 2012 16:30 EST

Shigenori Matsumoto

Source: Tohoku Math. J. (2), Volume 64, Number 4, 569--592.

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.

Automorphisms of an irregular surface of general type acting trivially in cohomology, II

Thu, 20 Dec 2012 16:30 EST

Jin-Xing Cai

Source: Tohoku Math. J. (2), Volume 64, Number 4, 593--605.

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.

Convexity of reflective submanifolds in symmetric $R$-spaces

Thu, 20 Dec 2012 16:30 EST

Peter Quast, Makiko Sumi Tanaka

Source: Tohoku Math. J. (2), Volume 64, Number 4, 607--616.

Abstract:
We show that every reflective submanifold of a symmetric $R$-space is (geodesically) convex.

Voronoi tilings hidden in crystals ---the case of maximal abelian coverings---

Mon, 08 Apr 2013 16:24 EDT

Tadao Oda

Source: Tohoku Math. J. (2), Volume 65, Number 1, 1--30.

Abstract:
Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadri as a by-product of Geometric Invariant Theory, he shows that the Voronoi tiling (also known as the Wigner-Seitz tiling) is hidden in the crystal, that is, the crystal does not intrude the interiors of the top-dimensional Voronoi cells. The result turns out to be closely related to the tropical Abel-Jacobi map of the associated compact tropical curve.

Hessian manifolds of nonpositive constant Hessian sectional curvature

Mon, 08 Apr 2013 16:24 EDT

Hitoshi Furuhata, Takashi Kurose

Source: Tohoku Math. J. (2), Volume 65, Number 1, 31--42.

Abstract:
We classify the maximal Hessian manifolds of constant Hessaian sectional curvature nonpositive.

Higher dimensional minimal submanifolds generalizing the catenoid and helicoid

Mon, 08 Apr 2013 16:24 EDT

Jaigyoung Choe, Jens Hoppe

Source: Tohoku Math. J. (2), Volume 65, Number 1, 43--55.

Abstract:
For each $k$-dimensional complete minimal submanifold $M$ of $\boldsymbol{S}^n$ we construct a $(k+1)$-dimensional complete minimal immersion of $M\times \boldsymbol{R}$ into $\boldsymbol{R}^{n+2}$ and $(k+1)$-dimensional minimal immersions of $M\times \boldsymbol{R}$ into $\boldsymbol{R}^{2n+3},\boldsymbol{H}^{2n+3}$ and $\boldsymbol{S}^{2n+3}$. Also from the Clifford torus $M=\boldsymbol{S}^{k}(1/\sqrt{2})\times\boldsymbol{S}^{k}(1/\sqrt{2})$ we construct a $(2k+2)$-dimensional complete minimal helicoid in \boldsymbol{R}^{2k+3}$.

Localization for an Anderson-Bernoulli model with generic interaction potential

Mon, 08 Apr 2013 16:24 EDT

Hakim Boumaza

Source: Tohoku Math. J. (2), Volume 65, Number 1, 57--74.

Abstract:
We present a result of localization for a matrix-valued Anderson-Bernoulli operator acting on the space of $\boldsymbol{C}^N$-valued square-integrable functions, for an arbitrary $N$ larger than 1, whose interaction potential is generic in the real symmetric matrices. For such a generic real symmetric matrix, we construct an explicit interval of energies on which we prove localization, in both spectral and dynamical senses, away from a finite set of critical energies. This construction is based upon the formalism of the Fürstenberg group to which we apply a general criterion of density in semisimple Lie groups. The algebraic nature of the objects we are considering allows us to prove a generic result on the interaction potential and the finiteness of the set of critical energies.

Weighted maximal inequalities for martingales

Mon, 08 Apr 2013 16:24 EDT

Adam Osȩkowski

Source: Tohoku Math. J. (2), Volume 65, Number 1, 75--91.

Abstract:
The paper contains the study of sharp weighted versions of the classical Doob's weak-type estimates for real-valued martingales. As a by-product, some results concerning the structure of Muckenhoupt's classes are obtained. The proof rests on Bellman function method, i.e., it is based on the construction of a special function having appropriate concavity and majorization properties.

On the ACC for lengths of extremal rays

Mon, 08 Apr 2013 16:24 EDT

Osamu Fujino, Yasuhiro Ishitsuka

Source: Tohoku Math. J. (2), Volume 65, Number 1, 93--103.

Abstract:
We discuss the ascending chain condition for lengths of extremal rays. We prove that the lengths of extremal rays of $n$-dimensional $\boldsymbol{Q}$-factorial toric Fano varieties with Picard number one satisfy the ascending chain condition.

Normal singularities with torus actions

Mon, 08 Apr 2013 16:24 EDT

Alvaro Liendo, Hendrik Süss

Source: Tohoku Math. J. (2), Volume 65, Number 1, 105--130.

Abstract:
We propose a method to compute a desingularization of a normal affine variety $X$ endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of $X$. In particular, we give criteria for $X$ to have only rational, ($\boldsymbol{Q}$-)factorial, or ($\boldsymbol{Q}$-)Gorenstein singularities. We also give partial criteria for $X$ to be Cohen-Macaulay or log-terminal. Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varieties in the case where the action is of complexity one.

A comparison theorem for Steiner minimum trees in surfaces with curvature bounded below

Mon, 08 Apr 2013 16:24 EDT

Shintaro Naya, Nobuhiro Innami

Source: Tohoku Math. J. (2), Volume 65, Number 1, 131--157.

Abstract:
Let $D$ be a compact polygonal Alexandrov surface with curvature bounded below by $\kappa $. We study the minimum network problem of interconnecting the vertices of the boundary polygon $\partial D$ in $D$. We construct a smooth polygonal surface $\widetilde D$ with constant curvature $\kappa $ such that the length of its minimum spanning trees is equal to that of $D$ and the length of its Steiner minimum trees is less than or equal to $D$'s. As an application we show a comparison theorem of Steiner ratios for polygonal surfaces.