Showing papers in "Tohoku Mathematical Journal in 2013"

Normal singularities with torus actions

Abstract: We propose a method to compute a desingularization of a normal affine varietyX 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, (Q-)factorial, or (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.

Hardy type inequalities on balls

Abstract: Hardy type inequalities are presented on balls with radius $R$ at the origin in ${\boldsymbol R}^n$ with $n=2$ at least. A special attention is paid on the behavior of functions on the boundary.

On the Harnack inequality for parabolic minimizers in metric measure spaces

Abstract: In this note we consider problems related to parabolic partial dif- ferential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincare inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the sufficiency of the Grigor'yan-Saloff-Coste theorem for general p > 1 in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting.

Second variational formula and the stability of legendrian minimal submanifolds in sasakian manifolds

Abstract: In this paper, we investigate compact Legendrian submanifolds $L$ in Sasakian manifolds $M$, which have extremal volume under Legendrian deformations. We call such a submanifold $L$-minimal Legendrian submanifold. We derive the second variational formula for the volume of $L$ under Legendrian deformations in $M$. Applying this formula, we investigate the stability of $L$-minimal Legendrian curves in Sasakian space forms, and show the $L$-instability of $L$-minimal Legendrian submanifolds in $S^{2n+1}(1)$. Moreover, we give a construction of $L$-minimal Legendrian submanifolds in ${\boldsymbol R}^{2n+1}(-3)$.

Higher dimensional minimal submanifolds generalizing the catenoid and helicoid

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}$.

Submanifolds with constant scalar curvature in a unit sphere

Abstract: We study the submanifolds in the unit sphere ${\boldsymbol S}^{n+p}$ with constant scalar curvature and parallel normalized mean curvature vector field. In this case, we can generalize the work of the second author about hypersurfaces in Hypersurfaces with constant scalar curvature in space forms to submanifolds in a unit sphere.

Meromorphic continuations of local zeta functions and their applications to oscillating integrals

Abstract: We introduce a new method which enables us to calculate the coefficients of the poles of local zeta functions very precisely and prove some explicit formulas. Some vanishing theorems for the candidate poles of local zeta functions will be also given. Moreover we apply our method to oscillating integrals and obtain an explicit formula for the coefficients of their asymptotic expansions.

Non-existence of certain CM abelian varieties with prime power torsion

Abstract: In this paper, we study a conjecture of Rasmussen and Tamagawa, on the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion. Our result is related with abelian varieties which have complex multiplication over their fields of definition.

Einstein metrics and Yamabe invariants of weighted projective spaces

Abstract: An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.

Hessian manifolds of nonpositive constant hessian sectional curvature

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

Polyharmonic functions of infinite order on annular regions

Abstract: Polyharmonic functions $f$ of infinite order and type $\tau$ on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients $f_{k,l}(r)$ of a polyharmonic function $f$ of infinite order and type $0$ can be extended to analytic functions on the complex plane cut along the negative semiaxis. The second main result gives a constructive procedure via Fourier-Laplace series for the analytic extension of a polyharmonic function on annular region $A(r_{0},r_{1})$ of infinite order and type less than $1/2r_{1}$ to the kernel of the harmonicity hull of the annular region. The methods of proof depend on an extensive investigation of Taylor series with respect to linear differential operators with constant coefficients.

Weighted maximal inequalities for martingales

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.

New examples of sasaki-einstein manifolds

Abstract: In this note, stimulated by the existence result by Futaki, Ono and Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on $S^1$-bundles associated to canonical line bundles of ${\boldsymbol P}^1({\boldsymbol C})$-bundles over Kahler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here our examples include non-toric Sasaki-Einstein manifolds.

Regularized periods of automorphic forms on $gl(2)$

Abstract: In this paper, we study regularized periods of cusp forms and Eisenstein series on $GL(2)$ introduced by Masao Tsuzuki.

Some examples of self-similar solutions and translating solitons for Lagrangian mean curvature flow

Abstract: We construct examples of self-similar solutions and translating solitons for Lagrangian mean curvature flow by extending the method of Joyce, Lee and Tsui. Those examples include examples in which the Lagrangian angle is arbitrarily small as the examples of Joyce, Lee and Tsui. The examples are non-smooth zero-Maslov class Lagrangian self-expanders which are asymptotic to a pair of planes intersecting transversely.

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

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.

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

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.

Large deviations for symmetric stable processes with Feynman-Kac functionals and its application to pinned polymers

Abstract: Let $ u$ and $\mu$ be positive Radon measures on ${\boldsymbol R} ^d$ in Green-tight Kato class associated with a symmetric $\alpha$-stable process $(X_t , P_x)$ on ${\boldsymbol R}^d$, and $A_t ^ u$ and $A_t ^\mu$ the positive continuous additive functionals under the Revuz correspondence to $ u$ and $\mu$. For a non-negative $\beta$, let $P_{x,t} ^{\beta \mu}$ be the law $X_t$ weighted by the Feynman-Kac functional $\exp(\beta A_t ^\mu)$, i.e., $P_{x,t} ^\mu =(Z_{x,t} ^\mu)^{-1}\exp(\beta A_t ^\mu)P_x$, where $Z_{x,t} ^\mu$ is a normalizing constant. We show that $A_t ^ u /t$ obeys the large deviation principle under $P_{x,t}^{\beta \mu}$. We apply it to a polymer model to identify the critical value $\beta _{\rm cr}$ such that the polymer is pinned under the law $P^{\beta \mu} _{x,t} $ if and only if $\beta$ is greater than $\beta_{\rm cr}$. The value $\beta _{\rm cr} $ is characterized by the rate function.

Mahler measure and Weber's class number problem in the cyclotomic $\boldsymbol{Z}_p$-extension of $\boldsymbol{Q}$ for odd prime number $p$

Abstract: Let $p$ be a prime number and $n$ a non-negative integer. We denote by $h_{p, n}$ the class number of the $n$-th layer of the cyclotomic $\boldsymbol{Z}_p$-extension of $\boldsymbol{Q}$. Let $l$ be a prime number. In this paper, we assume that $p$ is odd and consider the $l$-divisibility of $h_{p,n}$. Let $f$ be the inertia degree of $l$ in the $p$-th cyclotomic field and $s$ the maximal exponent such that $p^s$ divides $l^{p-1}-1$. Set $r=\min\{n, s\}$. We define a certain explicit constant $G_{1}(p, r, f)$ in terms of the property of the residue class of $l$ modulo $p^r$. If $l$ is larger than $G_1(p, r, f)$, then the integer $h_{p, n}/h_{p, n-1}$ is coprime with $l$. Our proof refines Horie's method.

Streamlines concentration and application to the incompressible navier-stokes equations

Abstract: For a smooth domain containing the origin, we consider a divergence-free vector field and exclude certain types of possible isolated singularities at the origin, based on the geometry of streamlines that go near that possible singular point

A local signature for fibered 4-manifolds with a finite group action

Abstract: Let $p$ be a finite regular covering on a 2-sphere with at least three branch points. In this paper, we construct a local signature for the class of fibered 4-manifolds whose general fibers are isomorphic to the covering $p$.

K3 surfaces with an automorphism of order 11

Abstract: This paper concerns $K3$ surfaces with automorphisms of order 11 in arbitrary characteristic. Specifically we study the wild case and prove that a generic such surface in characteristic 11 has Picard number 2. We also construct $K3$ surfaces with an automorphism of order 11 in every characteristic, and supersingular $K3$ surfaces whenever possible.

Localization for an Anderson-Bernoulli model with generic interaction potential

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 Furstenberg 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.

Reflection arrangements are hereditarily free

Abstract: Suppose that W is a finite, unitary, reflection group acting on the complex vector space V . Let A = A(W ) be the associated hyperplane arrangement of W . Terao has shown that each such reflection arrangement A is free. Let L(A) be the intersection lattice of A. For a subspace X in L(A) we have the restricted arrangement A in X by means of restricting hyperplanes from A to X. In 1992, Orlik and Terao conjectured that each such restriction is again free. In this note we settle the outstanding cases confirming the conjecture. In 1992, Orlik and Terao also conjectured that every reflection arrangement is hereditarily inductively free. In contrast, this stronger conjecture is false however; we give two counterexamples.

On the two-variables main conjecture for extensions of imaginary quadratic fields

Abstract: Let $p$ be a prime number at least 5, and let $k$ be an imaginary quadratic number field in which $p$ decomposes into two conjugate primes. Let $k_\infty$ be the unique ${\boldsymbol Z}_p^2$-extension of $k$, and let $K_\infty$ be a finite extension of $k_\infty$, abelian over $k$. We prove that in $K_\infty$, the characteristic ideal of the projective limit of the $p$-class group coincides with the characteristic ideal of the projective limit of units modulo elliptic units. Our approach is based on Euler systems, which were first used in this context by Rubin.

On the ACC for lengths of extremal rays

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

The logarithmic growth of element of Robba ring which satisfies Frobenius equation over bounded Robba ring

Abstract: We study the logarithmic growth of an element of the Robba ring which satisfies a Frobenius equation over the bounded Robba ring. Chiarellotto and Tsuzuki computed the logarithmic growth of analytic functions on the open unit disc with coefficients in a $p$-adic local field which satisfy Frobenius equations over bounded functions of rank 2. We extend their result by replacing those functions by elements of the Robba ring which satisfy Frobenius equations over the bounded Robba ring. Moreover, we will see, in special cases, the zeros of these functions have some cyclicity and the logarithmic growth can be computed by the zeros of these function.

Visible actions on flag varieties of type c and a generalization of the cartan decomposition

Abstract: We give a generalization of the Cartan decomposition for connected compact Lie groups of type C motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A groups. Let $G$ be a compact simple Lie group of type C, $K$ a Chevalley--Weyl involution-fixed point subgroup and $L,H$ Levi subgroups. We firstly show that $G=LKH$ holds if and only if either Case I: $(G,H)$ and $(G,L)$ are both symmetric pairs or Case II: $L$ is a Levi subgroup of maximal dimension and $H$ is an arbitrary maximal Levi subgroup up to switch of $L,H$. This classification gives a visible action of $L$ on the generalized flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on the direct product of $G/L$ and $G/H$. Secondly, we find a generalized Cartan decomposition $G=LBH$ explicitly, where $B$ is a subset of $K$. An application to multiplicity-free theorems of representations is also discussed.