Osaka Journal of Mathematics Articles (Project Euclid)

Quotients of bounded homogeneous domains by cyclic groups

Thu, 05 Aug 2010 15:41 EDT

Christian Miebach

Source: Osaka J. Math., Volume 47, Number 2, 331--352.

Abstract:
Let $D$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $\varphi$ be an automorphism of $D$ which generates a discrete subgroup $\Gamma$ of $\Aut_{\mathcal{O}}(D)$. It is shown that the complex space $D/\Gamma$ is Stein.

Torsionfree dimension of modules and self-injective dimension of rings

Wed, 21 Mar 2012 09:40 EDT

Chonghui Huang, Zhaoyong Huang

Source: Osaka J. Math., Volume 49, Number 1, 21--35.

Abstract:
Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n \geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_{R}^{i}(M, R)=0$ for any $1 \leq i \leq n$. Then we investigate the relation between these properties and the self-injective dimension of $R$.

Asymptotic behavior of solutions for damped wave equations with non-convex convection term on the half line

Wed, 21 Mar 2012 09:40 EDT

Itsuko Hashimoto, Yoshihiro Ueda

Source: Osaka J. Math., Volume 49, Number 1, 37--52.

Abstract:
We study the asymptotic stability of nonlinear waves for damped wave equations with a convection term on the half line. In the case where the convection term satisfies the convex and sub-characteristic conditions, it is known by the work of Ueda [7] and Ueda--Nakamura--Kawashima [10] that the solution tends toward a stationary solution. In this paper, we prove that even for a quite wide class of the convection term, such a linear superposition of the stationary solution and the rarefaction wave is asymptotically stable. Moreover, in the case where the solution tends to the non-degenerate stationary wave, we derive that the time convergence rate is polynomially (resp. exponentially) fast if the initial perturbation decays polynomially (resp. exponentially) as $x \to \infty$. Our proofs are based on a technical $L^{2}$ weighted energy method.

Tame and wild degree functions

Wed, 21 Mar 2012 09:40 EDT

Daniel Daigle

Source: Osaka J. Math., Volume 49, Number 1, 53--80.

Abstract:
We give examples of degree functions $\deg\colon R \to M \cup \{-\infty\}$, where $R$ is $\mathbb{C}[X,Y]$ or $\mathbb{C}[X,Y,Z]$ and $M$ is $\mathbb{Z}$ or $\mathbb{N}$, whose behaviour with respect to $\mathbb{C}$-derivations $D\colon R \to R$ is pathological in the sense that $\{\deg(Dx) - \deg(x) \mid x \in R\setminus \{0\}\}$ is not bounded above. We also give several general results stating that such pathologies do not occur when the degree functions satisfy certain hypotheses.

On the structure of Stanley--Reisner rings associated to cyclic polytopes

Wed, 21 Mar 2012 09:40 EDT

Janko Böhm, Stavros Argyrios Papadakis

Source: Osaka J. Math., Volume 49, Number 1, 81--100.

Abstract:
We study the structure of Stanley--Reisner rings associated to cyclic polytopes, using ideas from unprojection theory. Consider the boundary simplicial complex $\Delta(d,m)$ of the $d$-dimensional cyclic polytope with $m$ vertices. We show how to express the Stanley--Reisner ring of $\Delta(d,m+1)$ in terms of the Stanley--Reisner rings of $\Delta(d,m)$ and $\Delta(d-2,m-1)$. As an application, we use the Kustin--Miller complex construction to identify the minimal graded free resolutions of these rings. In particular, we recover results of Schenzel, Terai and Hibi about their graded Betti numbers.

Second order type-changing equations for a scalar function on a plane

Wed, 21 Mar 2012 09:40 EDT

Takahiro Noda, Kazuhiro Shibuya

Source: Osaka J. Math., Volume 49, Number 1, 101--124.

Abstract:
In this paper, we consider type-changing equations for one unknown function of two variables by using the theory of differential systems. We give fundamental properties and provide a notion of geometric solutions from a viewpoint of contact geometry of second order. Moreover, we study the structure of associated overdetermined systems and obtain an existence condition of solutions of a special class which are called parabolic solutions of type-changing equations.

Totally geodesic submanifolds of regular Sasakian manifolds

Wed, 21 Mar 2012 09:40 EDT

Thomas Murphy

Source: Osaka J. Math., Volume 49, Number 1, 125--132.

Abstract:
Let $Q^{m+1}$ denote the family of regular Sasakian manifolds whose base manifold $M^{2m}$ is a compact symmetric space. We provide a classification of the totally geodesic submanifolds of $Q^{m+1}$ which are invariant, anti-invariant of maximal dimension or contact CR with respect to the Sasakian structure. Such submanifolds are closely related to complex and totally real totally geodesic submanifolds of the Hermitian symmetric space $M^{2m}$.

A new look at Condition A

Wed, 21 Mar 2012 09:40 EDT

Quo-Shin Chi

Source: Osaka J. Math., Volume 49, Number 1, 133--166.

Abstract:
Ozeki and Takeuchi [14] introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and Münzner to many more examples via the Clifford representations; we will refer to these examples of Ozeki and Takeuchi and of Ferus, Karcher and Münzner collectively as OT-FKM type throughout the paper. Dorfmeister and Neher [5] then employed isoparametric triple systems [3, 4], which are algebraic in nature, to prove that Condition A alone implies the isoparametric hypersurface is of OT-FKM type. Their proof for the case of multiplicity pairs $\{3, 4\}$ and $\{7, 8\}$ rests on a fairly involved algebraic classification result [9] about composition triples. In light of the classification [2] that leaves only the four exceptional multiplicity pairs $\{4, 5\}, \{3, 4\}, \{7, 8\}$ and $\{6, 9\}$ unsettled, it appears that Condition A may hold the key to the classification when the multiplicity pairs are $\{3, 4\}$ and $\{7, 8\}$. Thus Condition A deserves to be scrutinized and understood more thoroughly from different angles. In this paper, we give a fairly short and rather straightforward proof of the result of Dorfmeister and Neher, with emphasis on the multiplicity pairs $\{3, 4\}$ and $\{7, 8\}$, based on more geometric considerations. We make it explicit and apparent that the octonion algebra governs the underlying isoparametric structure.

On the constructions of free and locally standard $\mathbb{Z}_{2}$-torus actions on manifolds

Wed, 21 Mar 2012 09:40 EDT

Li Yu

Source: Osaka J. Math., Volume 49, Number 1, 167--193.

Abstract:
We introduce an elementary way of constructing principal $(\mathbb{Z}_{2})^{m}$-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard $(\mathbb{Z}_{2})^{m}$-actions on closed manifolds for all $m \geq 1$, and then give a general way to construct all such $(\mathbb{Z}_{2})^{m}$-actions from the orbit space. Some related topology problems are also studied.

Knotoids

Wed, 21 Mar 2012 09:40 EDT

Vladimir Turaev

Source: Osaka J. Math., Volume 49, Number 1, 195--223.

Abstract:
We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister moves applied away from the endpoints of the underlying segment. We show that knotoids in $S^{2}$ generalize knots in $S^{3}$ and study the semigroup of knotoids. We also discuss applications to knots and invariants of knotoids.

Maximal ideal cycles over normal surface singularities of Brieskorn type

Wed, 21 Mar 2012 09:40 EDT

Kazuhiro Konno, Daisuke Nagashima

Source: Osaka J. Math., Volume 49, Number 1, 225--245.

Abstract:
For normal two dimensional hypersurface singularities of Brieskorn type, concrete descriptions are given to both the fundamental cycle and the maximal ideal cycle on a star-shaped good resolution space. It is determined when these two cycles coincide.

Self-mapping degrees of 3-manifolds

Wed, 21 Mar 2012 09:40 EDT

Hongbin Sun, Shicheng Wang, Jianchun Wu, Hao Zheng

Source: Osaka J. Math., Volume 49, Number 1, 247--269.

Abstract:
For each closed oriented $3$-manifold $M$ in Thurston's picture, the set of degrees of self-maps on $M$ is given.

The commutativity of Galois groups of the maximal unramified pro-$p$-extensions over the cyclotomic $\mathbb{Z}_{p}$-extensions II

Wed, 20 Jun 2012 09:12 EDT

Keiji Okano

Source: Osaka J. Math., Volume 49, Number 2, 271--295.

Abstract:
Let $p$ be an odd prime number and $K_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of a Galois $p$-extension $K$ over an imaginary quadratic field. We consider the Galois group $\tilde{X}(K_{\infty})$ of the maximal unramified pro-$p$-extension of $K_{\infty}$. In this paper, under certain assumptions, we give certain $K$ such that $\tilde{X}(K_{\infty})$ is abelian. Also, we give an example such that a special value of the characteristic polynomial of the Iwasawa module of $K_{\infty}$ determines whether $\tilde{X}(K_{\infty})$ is abelian or not.

Generalized energy conservation for Klein--Gordon type equations

Wed, 20 Jun 2012 09:12 EDT

Christiane Böhme, Fumihiko Hirosawa

Source: Osaka J. Math., Volume 49, Number 2, 297--323.

Abstract:
The aim of this paper is to derive energy estimates for solutions of the Cauchy problem for the Klein--Gordon type equation $u_{tt} - \bigtriangleup u + m(t)^{2} u = 0$. The coefficient $m$ is given by $m(t)^{2} = \lambda(t)^{2} + p(t)$ with a decreasing, smooth shape function $\lambda$ and an oscillating, smooth and bounded perturbation function $p$. We study under which assumptions for $\lambda$ and $p$ one can expect results about a generalization of energy conservation. The main theorems of this note deal with $m$ belonging to $C^{M}$, $M \ge 2$, and $m$ belonging to the Gevrey class $\gamma^{(s)}$, $s \ge 1$.

Fusion systems on metacyclic 2-groups

Wed, 20 Jun 2012 09:12 EDT

Benjamin Sambale

Source: Osaka J. Math., Volume 49, Number 2, 325--329.

Abstract:
Let $P$ be a finite metacyclic $2$-group and $\mathcal{F}$ a fusion system on $P$. We prove that $\mathcal{F}$ is nilpotent unless $P$ has maximal class or $P$ is homocyclic, i.e. $P$ is a direct product of two isomorphic cyclic groups. As a consequence we obtain the numerical invariants for $2$-blocks with metacyclic defect groups. This paper is a part of the author's PhD thesis.

Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system

Wed, 20 Jun 2012 09:12 EDT

Kenji Nishihara

Source: Osaka J. Math., Volume 49, Number 2, 331--348.

Abstract:
Consider the Cauchy problem for a system of weakly coupled heat equations, whose typical one is \begin{equation*} \left\{ \begin{array}{@{}ll@{}} u_{t} - \Delta u = \lvert v \rvert^{p-1}v,\\ v_{t} - \Delta v = \lvert u \rvert^{q-1}u, & (t, x) \in \mathbf{R}_{+} \times \mathbf{R}^{N}, \end{array} \right. \end{equation*} with $p,q \ge 1$, $pq > 1$. When $p,q$ satisfy $\max((p+1)/(pq-1),(q+1)/(pq-1)) < N/2$, the exponents $p,q$ are supercritical. In this paper we assort the supercritical exponent case to two cases. In one case both $p$ and $q$ are bigger than the Fujita exponent $\rho_{F}(N)=1+2/N$, while in the other case $\rho_{F}(N)$ is between $p$ and $q$. In both cases we obtain the time-global and unique existence of solutions for small data and their asymptotic behaviors. These observation will be applied to the corresponding system of the damped wave equations in low dimensional space.

Comparing hyperbolic distance with Kra's distance on the unit disk

Wed, 20 Jun 2012 09:12 EDT

Guowu Yao

Source: Osaka J. Math., Volume 49, Number 2, 349--356.

Abstract:
In this paper, Kra's distance $d_{K}$ and the hyperbolic distance $d_{\mathbb{D}}$ are compared on the unit disk $\mathbb{D}$. It is shown that $2d_{K} < d_{\mathbb{D}} < (\pi^{2}/8)\exp{d_{K}}$ on $\mathbb{D} \times \mathbb{D} \setminus \{\text{diagonal}\}$, where the constants $2$ and $\pi^{2}/8$ are sharp. As a consequence, this result gives a negative answer to a question posed by Martin [7] in a stronger sense.

Curvature properties of the slowness surface of the system of crystal acoustics for cubic crystals II

Wed, 20 Jun 2012 09:12 EDT

Otto Liess, Tetsuya Sonobe

Source: Osaka J. Math., Volume 49, Number 2, 357--391.

Abstract:
In this paper we study geometric properties of the slowness surface of the system of crystal acoustics for cubic crystals in the special case when the stiffness constants satisfy the condition $a = -2b$. The paper is a natural continuation of the paper [9] in which related properties were studied for general constants $a$ and $b$, but assuming that we were in the nearly isotropic case, in which case $a - b$ has to be small. We also take this opportunity to correct a statement made in [9]: see Remark 1.3.

A statistical relation of roots of a polynomial in different local fields III

Wed, 20 Jun 2012 09:12 EDT

Yoshiyuki Kitaoka

Source: Osaka J. Math., Volume 49, Number 2, 393--420.

Abstract:
Let $f(x)$ be a monic polynomial in $\mathbb{Z}[x]$. We have observed a statistical relation of roots of $f(x) \bmod p$ for different primes $p$, where $f(x)$ decomposes completely modulo $p$. We could guess what happens if $f(x)$ is irreducible and has at most one decomposition $f(x) = g(h(x))$ such that $g,h$ are monic polynomials over $\mathbb{Z}$ with $h(0) = 0$, $1 < \deg h < \deg f$. In this paper, we study cases that $f$ has two different such decompositions. Besides, we construct a series of polynomials f which have two non-trivial different decompositions $f(x) = g(h(x))$.

Pathwise uniqueness for singular SDEs driven by stable processes

Wed, 20 Jun 2012 09:12 EDT

Enrico Priola

Source: Osaka J. Math., Volume 49, Number 2, 421--447.

Abstract:
We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable Lévy processes with values in $\mathbb{R}^{d}$ having a bounded and $\beta$-Hölder continuous drift term. We assume $\beta > 1 - \alpha/2$ and $\alpha \in [1, 2)$. The proof requires analytic regularity results for the associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.

Generalised spin structures on 2-dimensional orbifolds

Wed, 20 Jun 2012 09:12 EDT

Hansjörg Geiges, Jesús Gonzalo Pérez

Source: Osaka J. Math., Volume 49, Number 2, 449--470.

Abstract:
Generalised spin structures, or $r$-spin structures, on a $2$-dimensional orbifold $\Sigma$ are $r$-fold fibrewise connected coverings (also called $r$\textsuperscript{th} roots) of its unit tangent bundle $ST\Sigma$. We investigate such structures on hyperbolic orbifolds. The conditions on $r$ for such structures to exist are given. The action of the diffeomorphism group of $\Sigma$ on the set of $r$-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the $3$-dimensional geometry $\widetilde{\mathrm{SL}}_{2}$.

Rack module enhancements of counting invariants

Wed, 20 Jun 2012 09:12 EDT

Aaron Haas, Garret Heckel, Sam Nelson, Jonah Yuen, Qingcheng Zhang

Source: Osaka J. Math., Volume 49, Number 2, 471--488.

Abstract:
We introduce a modified rack algebra $\mathbb{Z}[X]$ for racks $X$ with finite rack rank $N$. We use representations of $\mathbb{Z}[X]$ into rings, known as rack modules , to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide computations and examples to show that the new invariants are strictly stronger than the unenhanced counting invariant and are not determined by the Jones or Alexander polynomials.

Invariant complex structures on tangent and cotangent Lie groups of dimension six

Wed, 20 Jun 2012 09:12 EDT

Rutwig Campoamor-Stursberg, Gabriela P. Ovando

Source: Osaka J. Math., Volume 49, Number 2, 489--513.

Abstract:
This paper deals with left invariant complex structures on simply connected Lie groups, the Lie algebra of which is of the type $\mathrm{T}_{\pi} \mathfrak{h}=\mathfrak{h} \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\mathrm{T}_{\pi} \mathfrak{h}$ for $\mathfrak{h}$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constraint $J\mathfrak{h} = V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non-singular derivations of $\mathfrak{h}$. This fact allows different kinds of applications.

Filtered cohomological rigidity of bott towers

Wed, 20 Jun 2012 09:12 EDT

Hiroaki Ishida

Source: Osaka J. Math., Volume 49, Number 2, 515--522.

Abstract:
A Bott tower is an iterated $\mathbb{C}\mathrm{P}^{1}$-bundle over a point, where each $\mathbb{C}\mathrm{P}^{1}$-bundle is the projectivization of a rank $2$ decomposable complex vector bundle. For a Bott tower, the filtered cohomology is naturally defined. We show that isomorphism classes of Bott towers are distinguished by their filtered cohomology rings. We even show that any filtered cohomology ring isomorphism between two Bott towers is induced by an isomorphism of the Bott towers.

The Unknotting number and band-unknotting number of a knot

Wed, 20 Jun 2012 09:12 EDT

Tetsuya Abe, Ryo Hanaki, Ryuji Higa

Source: Osaka J. Math., Volume 49, Number 2, 523--550.

Abstract:
We show some results on the unknotting number and the band-unknotting number. Taniyama characterized knots whose unknotting number is half the crossing number minus one. We show that if the unknotting number of a knot is half the crossing number minus two, then the knot is the figure-eight knot, a positive $3$-braid knot, a negative $3$-braid knot or the connected sum of a $(2,r)$-torus knot and a $(2,s)$-torus knot for some odd integers $r,s \neq \pm 1$. In particular, we show that it is a $3$-braid knot. Taniyama and Yasuhara showed that the band-unknotting number of a knot is less than or equal to half the crossing number of the knot under our notation. We show that the equality holds if and only if the knot is the trivial knot or the figure-eight knot.

On positive quaternionic Kähler manifolds with $b_{4} = 1$

Mon, 15 Oct 2012 09:09 EDT

Jin Hong Kim, Hee Kwon Lee

Source: Osaka J. Math., Volume 49, Number 3, 551--562.

Abstract:
Let $M$ be a positive quaternionic Kähler manifold of dimension $4m$. In earlier papers, Fang and the first author showed that if the symmetry rank is greater than or equal to $[m/2]+3$, then $M$ is isometric to $\mathbf{HP}^{m}$ or $\mathit{Gr}_{2}(\mathbf{C}^{m+2})$. The goal of this paper is to give a more refined classification result for positive quaternionic Kähler manifolds (in particular, of relatively low dimension or with even $m$) whose fourth Betti number equals one. To be precise, we show in this paper that if the symmetry rank of $M$ with $b_{4}(M)=1$ is no less than $[m/2]+2$ for $m \ge 5$, then $M$ is isometric to $\mathbf{HP}^{m}$.

Subelliptic estimates for overdetermined systems of quadratic differential operators

Mon, 15 Oct 2012 09:09 EDT

Karel Pravda-Starov

Source: Osaka J. Math., Volume 49, Number 3, 563--611.

Abstract:
We prove global subelliptic estimates for systems of quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous work, we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their Weyl symbols, called singular space, which rules a number of fairly general properties of non-elliptic quadratic operators. About the subelliptic properties of these operators, we established that quadratic operators with zero singular spaces fulfill global subelliptic estimates with a loss of derivatives depending on certain algebraic properties of the Hamilton maps associated to their Weyl symbols. The purpose of the present work is to prove similar global subelliptic estimates for overdetermined systems of quadratic operators. We establish here a simple criterion for the subellipticity of these systems giving an explicit measure of the loss of derivatives and highlighting the non-trivial interactions played by the different operators composing those systems.

Real characters in blocks

Mon, 15 Oct 2012 09:09 EDT

Laszlo Héthelyi, Erzsebet Horváth, Endre Szabó

Source: Osaka J. Math., Volume 49, Number 3, 613--623.

Abstract:
We consider real versions of Brauer's $\mathrm{k}(B)$ conjecture, Olsson's conjecture and Eaton's conjecture. We prove the real version of Eaton's conjecture for $2$-blocks of groups with cyclic defect group and for the principal $2$-blocks of groups with trivial real core. We also characterize $G$-classes, real and rational $G$-classes of the defect group of$B$.

On vector valued Siegel modular forms of degree 2 with small levels

Mon, 15 Oct 2012 09:09 EDT

Hiroki Aoki

Source: Osaka J. Math., Volume 49, Number 3, 625--651.

Abstract:
In this paper, we show that the space of vector valued Siegel modular forms of $\Gamma_{0} (N) \subset \mathrm{Sp}(2, \mathbb{Z})$ with respect to the symmetric tensor of degree $2$ has a simple unified structure for $N=2,3,4$. Each structure is similar to the structure of the full modular group.

Bubbletons are not embedded

Mon, 15 Oct 2012 09:09 EDT

Martin Kilian

Source: Osaka J. Math., Volume 49, Number 3, 653--663.

Abstract:
We discuss constant mean curvature bubbletons in Euclidean 3-space via dressing with simple factors, and prove that single-bubbletons are not embedded.

On the singular impulsive functional integral equations involving nonlocal conditions

Mon, 15 Oct 2012 09:09 EDT

Rong-Nian Wang, Yan Wang, De-Han Chen

Source: Osaka J. Math., Volume 49, Number 3, 665--685.

Abstract:
In this paper, of concern is the singular impulsive functional integral equations subject to nonlocal conditions in a Banach space. Sufficient conditions, ensuring the existence of solutions, are presented. An example is also given to illustrate the applications of the abstract results. Our results essentially extend some existing results in this area.

Zeta function of degenerate plane curve singularity

Mon, 15 Oct 2012 09:09 EDT

Lê Quy Thuong

Source: Osaka J. Math., Volume 49, Number 3, 687--697.

Abstract:
We introduce in this paper a new resolution graph for an isolated complex plane curve singularity and then calculate the monodromy zeta function and the Alexander polynomial for the singularity in terms of this graph.

Multiple solutions for superlinear $p$-Laplacian Neumann problems

Mon, 15 Oct 2012 09:09 EDT

Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu

Source: Osaka J. Math., Volume 49, Number 3, 699--740.

Abstract:
Our main goal is to prove the existence of multiple solutions with precise sign information for a Neumann problem driven by the $p$-Laplacian differential operator with a ($p-1$)-superlinear term which does not satisfy the Ambrosetti--Rabinowitz condition. Using minimax methods we show that the problem has five nontrivial smooth solutions, two positive, two negative and the fifth nodal. In the semilinear case ($p = 2$), using Morse theory, we produce a second nodal solution (for a total of six nontrivial smooth solutions).

Twisted cohomology for hyperbolic three manifolds

Mon, 15 Oct 2012 09:09 EDT

Pere Menal-Ferrer, Joan Porti

Source: Osaka J. Math., Volume 49, Number 3, 741--769.

Abstract:
For a complete hyperbolic three manifold $M$, we consider the representations of $\pi_{1}(M)$ obtained by composing a lift of the holonomy with complex finite dimensional representations of $\mathrm{SL}(2,\mathbf{C})$. We prove a vanishing result for the cohomology of $M$ with coefficients twisted by these representations, using techniques of Matsushima--Murakami. We give some applications to local rigidity.

Translation planes of odd order via Dembowski--Ostrom polynomials

Mon, 15 Oct 2012 09:09 EDT

Ulrich Dempwolff, Peter Müller

Source: Osaka J. Math., Volume 49, Number 3, 771--794.

Abstract:
We describe a class of translation planes whose orders are of the form $q^{n}$, where $n$ is odd and $q$ is an odd prime power $>3$. These planes have the property that a translation complement fixes a triangle and acts transitively on the set of non-vertices of each side. The planes form an odd order analogue to the planes of Kantor--Williams [17] which have even order. The construction of the planes is based on a certain type of Dembowski--Ostrom polynomials.

On deformations of generalized Calabi--Yau and generalized $\mathrm{SU}(n)$-structures

Mon, 15 Oct 2012 09:09 EDT

Ryushi Goto

Source: Osaka J. Math., Volume 49, Number 3, 795--832.

Abstract:
In this paper we will introduce the new notion of generalized geometric structures defined by systems of closed differential forms. From a cohomological point of view, we develop a unified approach to deformation problems and establish a criterion for unobstructed deformations of the generalized geometric structures. We construct the moduli spaces of the structures with the action of $d$-closed $b$-fields and show that the period map of the moduli space is locally injective under the certain cohomological condition (the local Torelli type theorem). We apply our approach to generalized Calabi--Yau structures and generalized $\mathrm{SU}(n)$-structures and obtain unobstructed deformations of generalized Calabi--Yau structures if the $dd^{\mathcal{J}}$-property is satisfied. We also have unobstructed deformations of generalized $\mathrm{SU}(n)$-structures and show that the period map of the moduli space of generalized $\mathrm{SU}(n)$-structures is locally injective.

A transcendental approach to Kollár's injectivity theorem

Mon, 15 Oct 2012 09:09 EDT

Osamu Fujino

Source: Osaka J. Math., Volume 49, Number 3, 833--852.

Abstract:
We treat Kollár's injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Kollár type cohomology injectivity theorems. Our main theorem is formulated for a compact Kähler manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete Kähler metric. We need neither covering tricks, desingularizations, nor Leray's spectral sequence.

The expected volume and surface area of the Wiener sausage in odd dimensions

Wed, 19 Dec 2012 09:21 EST

Yuji Hamana

Source: Osaka J. Math., Volume 49, Number 4, 853--868.

Abstract:
We consider the Wiener sausage up to time $t$ associated with a closed ball. A formula for the expected volume of the Wiener sausage is obtained in odd dimensions. In these cases, we also find that the formula leads to the asymptotic expansion for large $t$ and each coefficient is represented by zeros of a modified Bessel function of the second kind. Moreover we obtain a formula for the expected surface area of the Wiener sausage.

Simple proofs of some theorems in block theory of finite groups

Wed, 19 Dec 2012 09:21 EST

Masafumi Murai

Source: Osaka J. Math., Volume 49, Number 4, 869--873.

Abstract:
We give simple proofs of Laradji's theorem on blocks with central defect groups, Watanabe's theorem on the Glauberman--Watanabe correspondences of blocks and Robinson's theorem on defect groups of $p$-blocks of $p$-solvable groups attaining Brauer's upper bound for the number of irreducible characters.

Unknotting the spun $T^{2}$-knot of a classical torus knot

Wed, 19 Dec 2012 09:21 EST

Inasa Nakamura

Source: Osaka J. Math., Volume 49, Number 4, 875--899.

Abstract:
We show that for the closure of a classical braid which satisfies certain conditions, the spun $T^{2}$-knot of the classical knot has the unknotting number one. This gives an alternative proof of the fact that the spun $T^{2}$-knot of a classical torus knot has the unknotting number one.

Discreteness criterion in $\mathrm{SL}(2,\mathbb{C})$ by a test map

Wed, 19 Dec 2012 09:21 EST

Wensheng Cao

Source: Osaka J. Math., Volume 49, Number 4, 901--907.

Abstract:
In the paper [12], Yang conjectured that a nonelementary subgroup $G$ of $\mathrm{SL}(2, \mathbb{C})$ containing elliptic elements is discrete if for each elliptic element $g \in G$ the group $\langle f, g \rangle$ is discrete, where $f \in \mathrm{SL}(2,\mathbb{C})$ is a test map being loxodromic or elliptic. By embedding $\mathrm{SL}(2,\mathbb{C})$ into $\mathrm{U}(1,1; \mathbb{H})$, we give an affirmative answer to this question. As an application, we show that a nonelementary and nondiscrete subgroup of $\mathrm{Isom}(H^{3})$ must contain an elliptic element of order at least 3.

On charts with two crossings II

Wed, 19 Dec 2012 09:21 EST

Teruo Nagase, Akiko Shima

Source: Osaka J. Math., Volume 49, Number 4, 909--929.

Abstract:
Let $\Gamma$ be a chart with at most two crossings. In this paper, we show that if $\Gamma$ is a 2-minimal generalized $n$-chart with $n \ge 5$, then $\Gamma$ contains at least $4n-10$ black vertices. And we show that if the closure of the surface braid represented by $\Gamma$ is a disjoint union of spheres, then $\Gamma$ is a ribbon chart. Hence the closure is a ribbon surface.

Minimal surfaces of genus one with catenoidal ends

Wed, 19 Dec 2012 09:21 EST

Shin Kato, Hisayoshi Muroya

Source: Osaka J. Math., Volume 49, Number 4, 931--992.

Abstract:
We give a necessary and sufficient condition for the existence of an $n$-end catenoid of genus one with prescribed flux. By using the condition, we construct new examples of families whose flux data go near to that of ``the catenoid of genus one''.

On the distribution of $\alpha p$ modulo one for primes $p$ of a special form

Wed, 19 Dec 2012 09:21 EST

San-Ying Shi

Source: Osaka J. Math., Volume 49, Number 4, 993--1004.

Abstract:
In this paper it is proved that for any irrational $\alpha$ and some $0 < \theta \le 1.5/100$, there are infinitely many primes $p$ such that $p+2$ has at most two prime factors and $\lVert\alpha p+\beta\rVert < p^{-\theta}$ which improves K. Matomäki's result $\theta < 1/1000$.

Stability conditions and $\mu$-stable sheaves on K3 surfaces with Picard number one

Wed, 19 Dec 2012 09:21 EST

Kotaro Kawatani

Source: Osaka J. Math., Volume 49, Number 4, 1005--1034.

Abstract:
In this article, we show that some semi-rigid $\mu$-stable sheaves on a projective K3 surface $X$ with Picard number $1$ are stable under Bridgeland's stability condition. As a consequence of our work, we show that the special set $U(X) \subset \mathrm{Stab}(X)$ introduced by Bridgeland reconstructs $X$ itself. This gives a sharp contrast to the case of an abelian surface.

The second variational formula of the $k$-energy and $k$-harmonic curves

Wed, 19 Dec 2012 09:21 EST

Shun Maeta

Source: Osaka J. Math., Volume 49, Number 4, 1035--1063.

Abstract:
In [4], J. Eells and L. Lemaire introduced $k$-energy and $k$-harmonic maps. In 1989, S.B. Wang [17] showed the first variation formula of the $k$-energy. In this paper, we give the second variation formula of $k$-energy and a notion of weakly stable and unstable. We also study $k$-harmonic maps into product Riemannian manifolds and $k$-harmonic curves into Riemannian manifolds with constant sectional curvature. Moreover, we give some non-trivial solutions of $3$-harmonic curves.

Note on lower bounds of energy growth for solutions to wave equations

Wed, 19 Dec 2012 09:21 EST

Shin-ichi Doi, Tatsuo Nishitani, Hideo Ueda

Source: Osaka J. Math., Volume 49, Number 4, 1065--1085.

Abstract:
In this note we study lower bounds of energy growth for solutions to wave equations which are compact in space perturbations of the wave equation $\partial_{t}^{2}u - \Delta u = 0$. Assuming that there exists a null bicharacteristic $(x(t),\xi(t))$, parametrized by the time $t$, such that $x(t)$ remains inside a ball and $\xi(t)$ outside a ball for $t \geq 0$ we prove that the solution operator $R(t)$ is bounded from below by constant times $\sqrt{\lvert\xi(t)\rvert/\lvert\xi(0)\rvert}$ in the operator norm. We apply this result to examples constructed by the same idea as in Colombini and Rauch [1] and show that there exist compact in space perturbations which cause $\exp(ct^{\alpha})$ growth of the energy for any given $0\leq \alpha \leq 1$.

Locally conformal Kähler structures on compact solvmanifolds

Wed, 19 Dec 2012 09:21 EST

Hiroshi Sawai

Source: Osaka J. Math., Volume 49, Number 4, 1087--1102.

Abstract:
Let $(M, g, J)$ be a compact Hermitian manifold and $\Omega$ the fundamental 2-form of $(g, J)$. A Hermitian manifold $(M, g, J)$ is said to be locally conformal Kähler if there exists a closed 1-form $\omega$ such that $d\Omega=\omega \wedge \Omega$. The purpose of this paper is to investigate a relation between a locally conformal Kähler structure and the adapted differential operator on compact solvmanifolds.

Asymptotic profile of quenching in a system of heat equations coupled at the boundary

Wed, 19 Dec 2012 09:21 EST

Zhengce Zhang, Yanyan Li

Source: Osaka J. Math., Volume 49, Number 4, 1103--1119.

Abstract:
We study finite time quenching for the radial solutions of a system of heat equations coupled at the boundary condition. This system exhibits simultaneous and non-simultaneous quenching. In particular, three kinds of simultaneous quenching profiles are obtained for different nonlinear exponent regions and appropriate initial data.

Conformally flat hypersurfaces with Bianchi-type Guichard net

Wed, 27 Mar 2013 09:20 EDT

Udo Hertrich-Jeromin, Yoshihiko Suyama

Source: Osaka J. Math., Volume 50, Number 1, 1--30.

Abstract:
We obtain a partial classification result for generic $3$-dimensional conformally flat hypersurfaces in the conformal $4$-sphere: explicit analytic data are obtained for conformally flat hypersurfaces with Bianchi-type canonical Guichard net. This is the first classification result for conformally flat hypersurfaces without additional symmetry. We discuss the curved flat associated family for conformally flat hypersurfaces and show that it descends to an associated family of conformally flat hypersurfaces. The associated family of conformally flat hypersurfaces with Bianchi-type Guichard net is investigated.

Random walks and Kuramochi boundaries of infinite networks

Wed, 27 Mar 2013 09:20 EDT

Atsushi Kasue

Source: Osaka J. Math., Volume 50, Number 1, 31--51.

Abstract:
In this paper, we study a connected non-parabolic, or transient, network compactified with the Kuramochi boundary, and show that the random walk converges almost surely to a random variable valued in the harmonic boundary, and a function of finite Dirichlet energy converges along the random walk to a random variable almost surely and in $L^{2}$. We also give integral representations of solutions of Poisson equations on the Kuramochi compactification.

Global monodromy modulo 5 of the quintic-mirror family

Wed, 27 Mar 2013 09:20 EDT

Kennichiro Shirakawa

Source: Osaka J. Math., Volume 50, Number 1, 53--60.

Abstract:
The quintic-mirror family is a well-known one-parameter family of Calabi--Yau threefolds. A complete description of the global monodromy group of this family is not yet known. In this paper, we give a presentation of the global monodromy group in the general linear group of degree 4 over the ring of integers modulo 5.

Singular $\mathbb{Q}$-homology planes of negative Kodaira dimension have smooth locus of non-general type

Wed, 27 Mar 2013 09:20 EDT

Karol Palka, Mariusz Koras

Source: Osaka J. Math., Volume 50, Number 1, 61--114.

Abstract:
We show that if a normal $\mathbb{Q}$-acyclic complex surface has negative Kodaira dimension then its smooth locus is not of general type. This generalizes an earlier result of Koras--Russell for contractible surfaces.

The moduli space of Catanese--Ciliberto--Ishida surfaces

Wed, 27 Mar 2013 09:20 EDT

Masa-Nori Ishida

Source: Osaka J. Math., Volume 50, Number 1, 115--133.

Abstract:
We determine the moduli space of the surfaces of general type studied by Catanese, Ciliberto and Hirotaka Ishida by using the family of Hesse cubic curves.

On the classification of homogeneous $2$-spheres in complex Grassmannians

Wed, 27 Mar 2013 09:20 EDT

Jie Fei, Xiaoxiang Jiao, Liang Xiao, Xiaowei Xu

Source: Osaka J. Math., Volume 50, Number 1, 135--152.

Abstract:
In this paper we discuss a classification problem of homogeneous 2-spheres in the complex Grassmann manifold $G(k + 1, n + 1)$ by theory of unitary representations of the 3-dimensional special unitary group $\mathit{SU}(2)$. First we observe that if an immersion $x\colon S^{2} \to G(k + 1, n + 1)$ is homogeneous, then its image $x(S^{2})$ is a 2-dimensional $\rho(\mathit{SU}(2))$-orbit in $G(k + 1, n + 1)$, where $\rho\colon \mathit{SU}(2) \to U(n + 1)$ is a unitary representation of $\mathit{SU}(2)$. Then we give a classification theorem of homogeneous 2-spheres in $G(k + 1, n + 1)$. As an application we describe explicitly all homogeneous 2-spheres in $G(2, 4)$. Also we mention about an example of non-homogeneous holomorphic 2-sphere with constant curvature in $G(2, 4)$.

Quasitoric manifolds homeomorphic to homogeneous spaces

Wed, 27 Mar 2013 09:20 EDT

Michael Wiemeler

Source: Osaka J. Math., Volume 50, Number 1, 153--160.

Abstract:
We present some classification results for quasitoric manifolds $M$ with $p_{1}(M) = -\sum a_{i}^{2}$ for some $a_{i}\in H^{2}(M)$ which admit an action of a compact connected Lie-group $G$ such that $\dim M/G \leq 1$. In contrast to Kuroki's work [7, 6] we do not require that the action of $G$ extends the torus action on $M$.

Antipodal sets of symmetric $R$-spaces

Wed, 27 Mar 2013 09:20 EDT

Makiko Sumi Tanaka, Hiroyuki Tasaki

Source: Osaka J. Math., Volume 50, Number 1, 161--169.

Abstract:
We show that antipodal sets of symmetric $R$-spaces have the following properties. Any antipodal set is included in a great antipodal set and any two great antipodal sets are congruent.

A generalization of the Ross--Thomas slope theory

Wed, 27 Mar 2013 09:20 EDT

Yuji Odaka

Source: Osaka J. Math., Volume 50, Number 1, 171--185.

Abstract:
We give a formula for the Donaldson--Futaki invariants of certain type of semi test configurations, which essentially generalizes the Ross--Thomas slope theory [28]. The positivity (resp. non-negativity) of those ``a priori special'' Donaldson--Futaki invariants implies K-stability (resp. K-semistability). As an application, we prove the K-(semi)stability of certain polarized varieties with semi-log-canonical singularities, which generalizes some results of [28].

Equivariant maps between complex Stiefel manifolds

Wed, 27 Mar 2013 09:20 EDT

Zoran Z. Petrović, Branislav I. Prvulović

Source: Osaka J. Math., Volume 50, Number 1, 187--196.

Abstract:
The canonical circle action on complex Stiefel manifolds is considered. The corresponding mod $p$ index for all primes $p$ is computed and some Borsuk--Ulam type theorems are proved.

Conformal symmetries of self-dual hyperbolic monopole metrics

Wed, 27 Mar 2013 09:20 EDT

Nobuhiro Honda, Jeff Viaclovsky

Source: Osaka J. Math., Volume 50, Number 1, 197--249.

Abstract:
We determine the group of conformal automorphisms of the self-dual metrics on $n \# \mathbb{CP}^{2}$ due to LeBrun for $n \geq 3$, and Poon for $n = 2$. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space $\mathcal{H}^{3}$ minus a finite number of points, called monopole points. We show that for $n \geq 3$, any conformal automorphism is a lift of an isometry of $\mathcal{H}^{3}$ which preserves the set of monopole points. Furthermore, we prove that for $n = 2$, such lifts form a subgroup of index $2$ in the full automorphism group, which we show to be a semi-direct product $(\mathrm{U}(1) \times \mathrm{U}(1)) \rtimes \mathrm{D}_{4}$, where $\mathrm{D}_{4}$ is the dihedral group of order $8$.

On automorphisms of Klein surfaces with invariant subsets

Wed, 27 Mar 2013 09:20 EDT

E. Bujalance, G. Gromadzki

Source: Osaka J. Math., Volume 50, Number 1, 251--269.

Abstract:
It is well known that a group of automorphisms $G$ of an unbordered Klein surface $X$ of topological genus $g\geq 2$ in the orientable case and $g\geq 3$ otherwise has at most $84(g-\varepsilon)$ elements, where $\varepsilon =1$ or $2$ respectively. In the middle of the fifties, Oikawa used the cardinality $k$ of a $G$-invariant subset to introduce the bound $\lvert G\rvert \leq 12(g-1)+6k$ in the orientable case. Much later, T. Arakawa has generalized this result, involving $s=2$ or $3$ such subsets and showing in addition that the bound for $s=3$ is sharp for infinitely many configurations. Here we improve the bound of Arakawa for $s=2$, showing in particular that the last is never attained. In both orientable and non-orientable case, we also find bounds for arbitrary $s$ and show their sharpness for infinitely many topological configurations. Using another well known theorem of Oikawa and the canonical Riemann double cover, we get similar results for bordered Klein surfaces.

On knots with icon surfaces

Wed, 27 Mar 2013 09:20 EDT

Mario Eudave-Muñoz

Source: Osaka J. Math., Volume 50, Number 1, 271--285.

Abstract:
An ICON surface is an incompressible compact orientable nonseparating surface properly embedded in a knot exterior. We show that for any odd positive number $n$, there exist plenty of knots whose exteriors $E$ contain an ICON surface $F$ with $\lvert \partial F\rvert =n$. We also show that our examples satisfy the $\mathbb{Z}$-conjecture, that is, $\pi_{1}(E/F)\cong \mathbb{Z}$.