Showing papers in "Tohoku Mathematical Journal in 2007"

On the Feller property of Dirichlet forms generated by pseudo differential operators

Abstract: We show that a large class of regular symmetric Dirichlet forms is generated by pseudo differential operators. We calculate the symbols which are closely related to the semimartingale characteristics (Levy system) of the associated stochastic processes. Using the symbol we obtain estimates for the mean sojourn time of the process for balls. These estimates and a perturbation argument enable us to prove Holder regularity of the resolvent and semigroup; this entails that the semigroup has the Feller property.

Parabolic Harnack Inequality for the Mixture of Brownian Motion and Stable Process

Abstract: Let $X$ be a mixture of independent Brownian motion and symmetric stable process. In this paper we establish sharp bounds for transition density of $X$, and prove a parabolic Harnack inequality for nonnegative parabolic functions of $X$.

On pairs of geometric foliations on a cross-cap

Abstract: We obtain the topological configurations of the lines of curvature, the asymptotic and characteristic curves on a cross-cap, in the domain of a parametrisation of this surface as well as on the surface itself.

Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics

Abstract: We first present an alternative derivation of a local Weierstrass representation for flat surfaces in the real hyperbolic three-space, $\mathbb{H}^3$, using as a starting point an old result due to Luigi Bianchi. We then prove the following: let $M\subset \mathbb{H}^3$ be a flat compact connected smooth surface with $\partial M eq \emptyset$, transversal to a foliation of $\mathbb{H}^3$ by horospheres. If, along $\partial M$, $M$ makes a constant angle with the leaves of the foliation, then $M$ is part of an equidistant surface to a geodesic orthogonal to the foliation. We also consider the caustic surface associated with a family of parallel flat surfaces and prove that the caustic of such a familyis also a flat surface (possibly with singularities). Finally, a rigidity result for flat surfaces with singularities and a geometrical application of Schwarz's reflection principle are shown.

The cut locus of a two-sphere of revolution and Toponogov's comparison theorem

Abstract: We determine the structure of the cut locus of a class of two-spheres of revolution, which includes all ellipsoids of revolution. Furthermore, we show that a subclass of this class gives a new model surface for Toponogov's comparison theorem.

A combinatorial version of the grothendieck conjecture

Abstract: We study the "combinatorial anabelian geometry" that governs the relationship between the dual semi-graph of a pointed stable curve and various associated profinite fundamental groups of the pointed stable curve. Although many results of this type have been obtained previously in various particular situations of interest under unnecessarily strong hypotheses, the goal of the present paper is to step back from such "typical situations of interest" and instead to consider this topic in the abstract -- a point of view which allows one to prove results of this type in much greater generality under very weak hypotheses.

Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds

Abstract: We determine the local structure of all pseudo-Riemannian manifolds of dimensions greater than 3 whose Weyl conformal tensor is parallel and has rank 1 when treated as an operator acting on exterior 2-forms at each point. If one fixes three discrete parameters: the dimension, the metric signature (with at least two minuses and at least two pluses), and a sign factor accounting for semidefiniteness of the Weyl tensor, then the local-isometry types of our metrics correspond bijectively to equivalence classes of surfaces with equiaffine projectively flat torsionfree connections; the latter equivalence relation is provided by unimodular affine local diffeomorphisms. The surface just mentioned arises, locally, as the leaf space of a codimension-two parallel distribution on the pseudo-Riemannian manifold in question, naturally associated with its metric. We construct examples showing that the leaves of this distribution may form a fibration with the base which is a closed surface of any prescribed diffeomorphic type. Our result also completes a local classification of pseudo-Riemannian metrics with parallel Weyl tensor that are neither conformally flat nor locally symmetric: for those among such metrics which are not Ricci-recurrent, the Weyl tensor has rank 1, and so they belong to the class discussed in the previous paragraph; on the other hand, the Ricci-recurrent ones have already been classified by the second author.

Weak del Pezzo surfaces with irregularity

Abstract: We construct normal del Pezzo surfaces, and regular weak del Pezzo surfaces as well, with positive irregularity $q>0$. This can happen only over nonperfect fields. The surfaces in question are twisted forms of nonnormal del Pezzo surfaces, which were classified by Reid. The twisting is with respect to the flat topology and infinitesimal group scheme actions. The twisted surfaces appear as generic fibers for Fano-Mori contractions on certain threefolds with only canonical singularities.

WEAK SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS OVER THE FIELD OF $p$-ADIC NUMBERS

Abstract: Study of stochastic differential equations on the field of $p$-adic numbers was initiated by the second author and has been developed by the first author, who proved several results for the $p$-adic case, similar to the theory of ordinary stochastic integral with respect to Levy processes on Euclidean spaces. In this article, we present an improved definition of a stochastic integral on the field and prove the joint (time and space) continuity of the local time for $p$-adic stable processes. Then we use the method of random time change to obtain sufficient conditions for the existence of a weak solution of a stochastic differential equation on the field, driven by the $p$-adic stable process, with a Borel measurable coefficient.

Certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rationals

Abstract: We study, for any prime number $p$, the triviality of certain primary components of the ideal class group of the $\boldsymbol{Z}_p$-extension over the rational field. Among others, we prove that if $p$ is $2$ or $3$ and $l$ is a prime number not congruent to $1$ or $-1$ modulo $2p^2$, then $l$ does not divide the class number of the cyclotomic field of $p^u$th roots of unity for any positive integer $u$.

Meromorphic extensions of a class of zeta functions for two-dimensional billiards without eclipse

Abstract: The main purpose of the present paper is to show that a class of dynamical zeta functions associated with the so-called two-dimensional open billiard without eclipse have meromorphic extensions to the half-plane consisting of all complex numbers whose real parts are greater than a certain negative number. As an application, we verify that the zeta function for the length spectrum of the corresponding billiard table has the same property.

Equivariant K-theory of smooth toric varieties

Abstract: We characterize the smooth toric varieties for which the Merkurjev spectral sequence, connecting equivariant and ordinary K-theory, degenerates. We find under which conditions on the support of the fan the $E^2$ terms of the spectral sequence are invariants by subdivisions of the fan. Assuming these conditions, we describe explicitly the $E^2$ terms, linking them to the reduced homology of the fan.

Simultaneous similarity, bounded generation and amenability

Abstract: We prove that a discrete group $G$ is amenable if and only if it is strongly unitarizable in the following sense: every unitarizable representation $\pi$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $\pi$. Analogously, a $C^*$-algebra $A$ is nuclear if and only if any bounded homomorphism $u: A\to B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $\xi$ in the von Neumann algebra generated by the range of $u$ such that $a\to \xi u(a) \xi^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations. We apply this to answer several questions left open in our previous work concerning the length $L(A\otimes_{\max} B)$ of the maximal tensor product $A\otimes_{\max} B$ of two unital $C^*$-algebras, when we consider its generation by the subalgebras $A\otimes 1$ and $1\otimes B$. We show that if $L(A\otimes_{\max} B)<\infty$ either for $B=B(\ell_2)$ or when $B$ is the $C^*$-algebra (either full or reduced) of a non-Abelian free group, then $A$ must be nuclear. We also show that $L(A\otimes_{\max} B)\le d$ if and only if the canonical quotient map from the unital free product $A\,{\ast}\, B$ onto $A\otimes_{\max} B$ remains a complete quotient map when restricted to the closed span of the words of length at most $d$.

The critical dimensions of Hamachi shifts

Abstract: In 1981, Hamachi introduced an interesting class of type III shifts. Since these are not measure-preserving, one cannot use metric entropy to study them. As a possible substitute, we give estimates for the upper and lower critical dimensions of Hamachi shifts. These invariants have previously been used by the authors to study odometer actions.

A rough multiple Marcinkiewicz integral along continuous surfaces

Abstract: By means of the method of block decompositions for kernel functions and some delicate estimates on Fourier transforms, the L p (R m × R n × R)-boundedness of the multiple Marcinkiewicz integral is established along a continuous surface with rough kernel for some p> 1. The condition on the integral kernel is the best possible for the L 2 -boundedness of the multiple Marcinkiewicz integral operator. 1. Introduction. Let R N (N = m or n), N ≥ 2, be the N-dimensional Euclidean space and S N−1 the unit sphere in R N. For nonzero points x ∈ R m and y ∈ R n ,w e denote x � = x/|x| and y � = y/|y| .F orm ≥ 2a ndn ≥ 2, let Ω(x � ,y � ) ∈ L 1 (S m−1 × S n−1 ) be a homogeneous function of degree zero satisfying

Primitive ideals of the ring of differential operators on an affine toric variety

Abstract: We show that the classification of $A$-hypergeometric systems and that of multi-graded simple modules (up to shift) over the ring of differential operators on an affine toric variety are the same. We then show that the set of multi-homogeneous primitive ideals of the ring of differential operators is finite. Furthermore, we give conditions for the algebra being simple.

Splitting density for lifting about discrete groups

Abstract: We study splitting densities of primitive elements of a discrete subgroup of a connected non-compact semisimple Lie group of real rank one with finite center in another larger such discrete subgroup. When the corresponding cover of such a locally symmetric negatively curved Riemannian manifold is regular, the densities can be easily obtained from the results due to Sarnak or Sunada. Our main interest is a case where the covering is not necessarily regular. Specifically, for the case of the modular group and its congruence subgroups, we determine the splitting densities explicitly. As an application, we study analytic properties of the zeta function defined by the Euler product over elements consisting of all primitive elements which satisfy a certain splitting law for a given lifting.

Spectral synthesis in the fourier algebra and the varopoulos algebra

Abstract: The objects of study in this paper are sets of spectral synthesis for the Fourier algebra $A(G)$ of a locally compact group and the Varopoulos algebra $V(G)$ of a compact group with respect to submodules of the dual space. Such sets of synthesis are characterized in terms of certain closed ideals. For a closed set in a closed subgroup $H$ of $G,$ the relations between these ideals in the Fourier algebras of $G$ and $H$ are obtained. The injection theorem for such sets of synthesis is then a consequence. For the Fourier algebra of the quotient modulo a compact subgroup, an inverse projection theorem is proved. For a compact group, a correspondence between submodules of the dual spaces of $A(G)$ and $V(G)$ is set up and this leads to a relation between the corresponding sets of synthesis.

Hamiltonian actions and homogeneous Lagrangian submanifolds

Abstract: We consider a connected symplectic manifold M acted on properly and in a Hamiltonian fashion by a connected Lie group G. Inspired to the recent paper [3], see also [12] and [24], we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is given and also we describe under which condition a Lagrangian orbit is isolated. If M is a compact Kahler manifold we give a necessary and sufficient condition to an isometric action admits a Lagrangian orbit. Then we investigate Lagrangian homogeneous submanifolds on the symplectic cut and on the symplectic reduction. As an application of our results, we give new examples of Lagrangian homogeneous submanifolds on the blow-up at one point of the projective space and on the weighted projective spaces. Finally, applying Proposition 3.7 that we may call Lagrangian slice theorem for group acting with a fixed point, we give new examples of Lagrangian homogeneous submanifolds on irreducible Hermitian symmetric spaces of compact and noncompact type.

A characterization of symmetric siegel domains by convexity of cayley transform images

Abstract: We show that a homogeneous Siegel domain is symmetric if and only if its Cayley transform image is convex. Moreover, this convexity forces the parameter of the Cayley transform to be a specific one, so that the Cayley transform coincides with the inverse of the Cayley transform introduced by Kor´anyi and Wolf.

Dominant rational maps in the category of log schemes

Abstract: Kobayashi-Ochiai's theorem states that the set of dominant rational maps from a complex variety to a complex variety of general type is finite. Kazuya Kato conjectured a similar result in the category of log schemes. Our main theorem of this paper is a solution to his conjecture.

Periodic travelling wave solutions of a curvature flow equation in the plane

Abstract: In the plane, we consider a curvature flow equation in heterogeneous media with periodic horizontal striations, the periodicity in space is expressed by periodic (in vertical direction) coefficients in the equation. We prove the existence and uniqueness of a curve which travels upward periodically with an average speed. At each time, the graph of the curve is a periodic undulating line at a finite distance from a straight line with a given inclination angle. We also show that the average speed depends on the inclination angle monotonously. Moreover, for homogenization problem as the spatial period tends to zero, we estimate the average speed by the inclination angle and some means of the periodic coefficients.

Asymptotically saturated toric algebras

Abstract: We show the finite generation of certain invariant graded algebras defined on toric weak log Fano fibrations. These are the toric version of FGA algebras, recently introduced by Shokurov in connections to the existence of flips.

Non-regular pseudo-differential operators on the weighted Triebel-Lizorkin spaces

Abstract: We consider certain non-regular pseudo-differential operators and study the question of their boundedness on the weighted Triebel-Lizorkin and Besov spaces.

On the Hodge structure of degenerating hypersurfaces in toric varieties

Abstract: We introduce an algebraic method for describing the Hodge filtration of degenerating hypersurfaces in projective toric varieties. For this purpose, we show some fundamental properties of logarithmic differential forms on proper equivariant morphisms of toric varieties.