Showing papers in "Tohoku Mathematical Journal in 2000"

Toward the classification of higher-dimensional toric Fano varieties

Abstract: The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano verieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite successions of equivariant blow-ups and blow-downs through nonsingular toric Fano varieties. Especially, we get a new method for the classification of nonsingular toric Fano varieties of dimension at most four. These methods are extended to the case of Gorenstein toric Fano varieties endowed with natural resolutions of singularities. Especially, we easily get a new method for the classification of Gorenstein toric Fano surfaces.

Lie sphere geometry and integrable systems

Abstract: Two basic Lie-invariant forms uniquely defining a generic (hyper)surface in Lie sphere geometry are introduced. Particularly interesting classes of surfaces associated with these invariants are considered. These are the diagonally cyclidic surfaces and the Lie-minimal surfaces, the latter being the extremals of the simplest Lie-invariant functional generalizing the Willmore functional in conformal geometry. Equations of motion of a special Lie sphere frame are derived, providing a convenient unified treatment of surfaces in Lie sphere geometry. In particular, for diagonally cyclidic surfaces this approach immediately implies the stationary modified Veselov-Novikov equation, while the case of Lie-minimal surfaces reduces in a certain limit to the integrable coupled Tzitzeica system. In the framework of the canonical correspondence between Hamiltonian systms of hydrodynamic type and hypersurfaces in Lie sphere geometry, it is pointed out that invariants of Lie-geometric hypersurfaces coincide with the reciprocal invariants of hydrodynamic type systems. Integrable evolutions of surfaces in Lie sphere geometry are introduced. This provides an interpretation of the simplest Lie-invariant functional as the first local conservation law of the (2+1)-dimensional modified Veselov-Novikov hierarchy. Parallels between Lie sphere geometry and projective differential geometry of surfaces are drawn in the conclusion.

Homogeneous fractional integrals on Hardy spaces

Abstract: Mapping properties for the homogeneous fractional integral operator $T_{{\mit \Omega},\alpha}$ on the Hardy spaces $H^p(R^n)$ are studied. Our results give the extension of Stein-Weiss and Taibleson-Weiss's results for the boundedness of the Riesz potential operator $I_{\alpha}$ on the Hardy spaces $H^p(R^n)$.

Lines of principal curvature around umbilics and whitney umbrellas

Abstract: In this paper is studied the configuration of lines of curvature near a Whitney umbrella which is the unique stable singularity for maps of surfaces into R 3. The pattern of such configuration is established and characterized in terms of the 3-jet of the map. The result is used to establish an expression for the Euler-Poincare characteristic in terms of the number of umbilics and umbrellas. 1. Introduction. The bending or curvature pattern of a smooth mapping a : M -» R3, where M is a compact oriented two dimensional manifold, will be represented here by singular points, Sa, at which the mapping has rank less than 2 and the bending can be re- garded to be infinite; the umbilic points, Ua, at which the bending is finite but equal in all directions: and by the family of lines of principal curvature T\,a and Ti,a defined on M \ (Ua U 4. They showed that Darbouxian umbilic points characterize those with local structurally stable configuration, under small C3 deformations of the surface. See also the work of Bruce and Fidal (B-F). A study of principal foliations near the set Sa of singular points, aiming to characterize their local stability, was carried out by Gutierrez and Sotomayor (GS2). To this end they gave two sufficient conditions, the first of which is the Whitney singularity condition for stability of mappings in the sense of Singularity Theory (G-G). However, in this paper, an erroneous

Some semi-Riemannian volume comparison theorems

Abstract: Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local version of the Gunther and Bishop theorems is given and applied.

Operator-valued martingale transforms

Abstract: We develop a general theory of martingale transform operators with operator-valued multiplying sequences. Applications are given to classical operators such as Doob's maximal function and the square function. Some geometric properties of the underlying Banach spaces are also considered.

On Brumer's family of {RM}-curves of genus two

Abstract: We reconstruct Brumer's family with 3-parameters of curves of genus two whose jacobian varieties admit a real multiplication of discriminant 5. Our method is based on the descent theory in geometric Galois theory which can be compared with a classical problem of Noether. Namely, we first construct a 3-parameter family of polynomials $f(X)$ of degree 6 whose Galois group is isomorphic to the alternating group $A_5$. Then we study the family of curves defined by $Y^2=f(X)$, showing that they are equivalent to Brumer's family. The real multiplication will be described in three distinct ways, i.e., by Humbert's modular equation, by Poncelet's pentagon, and by algebraic correspondences.

Timelike surfaces with constant mean curvature in Lorentz three-space

Abstract: A cyclic surface in the Lorentz-Minkowski three-space is one that is foliated by circles. We classify all maximal cyclic timelike surfaces in this space, obtaining different families of non-rotational maximal surfaces. When the mean curvature is a non-zero constant, we prove that if the surface is foliated by circles in parallel planes, then it must be rotational. In particular, we obtain all timelike surfaces of revolution with constant mean curvature.

Quadratic relations for confluent hypergeometric functions

Abstract: We present a theory of intersection on the complex projective line for homology and cohomology groups defined by connections which are regular or not. We apply this theory to confluent hypergeometric functions, and obtain, as an analogue of period relations, quadratic relations satisfied by confluent hypergeometric functions.

Minimal maps between the hyperbolic discs and generalized Gauss maps of maximal surfaces in the anti-de sitter 3-space

Abstract: Problems related to minimal maps are studied. In particular, we prove an existence result for the Dirichlet problem at infinity for minimal diffeomorphisms between the hyperbolic discs. We also give a representation formula for a minimal diffeomorphism between the hyperbolic discs by means of the generalized Gauss map of a complete maximal surface in the anti-de Sitter 3-space.

Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere

Abstract: Our primary object of this paper is to give a representation formula for a surface with prescribed mean curvature in the (metric) 3-sphere by means of a single component of the generalized Gauss map. For a CMC (constant mean curvature) surface, we derive another representation formula by means of the adjusted Gauss map. These formulas are spherical versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. Spin versions of them are obtained as well.

Extreme stability and almost periodicity in a discrete logistic equation

Abstract: A sufficient condition is obtained for the existence of a globally asymptotically stable positive almost periodic solution of a discrete logistic equation. The sufficient condition becomes a necessary and sufficient condition for the global asymptotic stability of the positive equilibrium of the corresponding autonomous equation.

Periodic solutions of convex neutral functional differential equations

Abstract: In this paper we consider the existence of periodic solutions of neutral functional differential equations. It has been proved that for convex neutral functional differential equations of $D$-operator type with finite (or infinite) delay and hyperneutral functional differential equations with finite delay, there is a periodic solution if and only if there is a bounded solution. The results proved by Massera, Chow and Makay are generalized.

Periodic solutions for dissipative-repulsive systems

Abstract: It is proved that every dissipative-repulsive periodic system admits a periodic solution, which is comparable with some well-known results due to Yoshizawa, Hale and Lopes, and Burton and Zhang for dissipative systems.

Estimates of the fundamental solution for magnetic schrödinger operators and their applications

Abstract: We study the magnetic Schrodinger operator $H$ on $R^n$, $n\geq3$. We assume that the electrical potential $V$ and the magnetic potential {\bf a} belong to a certain reverse Holder class, including the case that $V$ is a non-negative polynomial and the components of {\bf a} are polynomials. We show some estimates for operators of Schrodinger type by using estimates of the fundamental solution for $H$. In particular, we show that the operator $ abla^2(-\Delta+V)^{-1}$ is a Calderon-Zygmund operator.

Modular inequalities for the Calderón operator

Abstract: If $P,Q:[0,\infty)\to$ are increasing functions and $T$ is the Calderon operator defined on positive or decreasing functions, then optimal modular inequalities $\int P(Tf)\leq C\int Q(f)$ are proved. If $P=Q$, the condition on $P$ is both necessary and sufficient for the modular inequality. In addition, we establish general interpolation theorems for modular spaces.

Configurations of conics with many tacnodes

Abstract: We investigate configurations of conics in the projective plane which have the property that the number of tacnodes is equal or close to the upper bound obtained from the Miyaoka-Yau inequality. We show that for 5 conics there are exactly 3 configurations, including 2 new ones, achieving the maximum 17 tacnodes, and for 6 conics the maximum number of tacnodes is 22, which together with previous results implies that the Miyaoka-Yau bound can never be achieved.

Isometric deformations of flat tori in the 3-sphere with nonconstant mean curvature

Abstract: For an isometric immersion $f$ of a flat torus into the unit 3-sphere, we show that if the mean curvature of $f$ is not constant, then the immersion $f$ admits a nontrivial isometric deformation preserving the total mean curvature.

Norm inequalities for fractional integrals of Laguerre and Hermite expansions

Abstract: Suppose the fractional integration operator $I^{\sigma}$ is generated by the sequence $\{(k+1)^{-\sigma}\}$ in the setting of Laguerre and Hermite expansions. Then, via projection formulas, the problem of the norm boundedness of $I^{\sigma}$ is reduced to the well-known fractional integration on the half-line. A corresponding result with respect to the modified Hankel transform is derived and its connection with the Laguerre fractional integration is indicated.

The polyhedral Hodge number $h^{2,1}$ and vanishing of obstructions

Abstract: We prove a vanishing theorem for the Hodge number $h^{2,1}$ of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope. In particular, the vanishing theorem for $h^{2,1}$ implies that these deformations are unobstructed.

The braidings of mapping class groups and loop spaces

Abstract: The disjoint union of mapping class groups forms a braided monoidal category. We give an explicit expression of braidings in terms of both their actions on the fundamental group of the surface and the standard Dehn twists. This braided monoidal category gives rise to a double loop space. We prove that the action of little 2-cube operad does not extend to the action of little 3-cube operad by showing that the Browder operation induced by 2-cube operad action is nontrivial. A rather simple expression of Reshetikhin-Turaev representation is given for the sixteenth root of unity in terms of matrices with entries of complex numbers. We show by matrix calculation that this representation is symmetric with respect to the braid structure.

Une remarque sur le problème de Cauchy pour l'opérateur différentiel de partie principale à coefficients polynomiaux II

Abstract: In our preceding article, by applying the results of Bieberbach, Fatou and Picard, we have studied the domain of holomorphy of the solution of the Cauchy problem for the differential operator with coefficients of entire functions. In this article, by employing the results of the modular function and its ordinary differential equation, we give a remark on the domain of holomorphy of the solution of the Cauchy problem for the differential operator of principal part with polynomial coefficients.

The dimension of the moduli space of superminimal surfaces of a fixed degree and conformal structure in the 4-sphere

Abstract: It was established by X. Mo and the author that the dimension of each irreducible component of the moduli space $\mathcal{M}_{d,g}(X)$ of branched superminimal immersions of degree $d$ from a Riemann surface $X$ of genus $g$ into $C P^3$ lay between $2d-4g+4$ and $2d-g+4$ for $d$ sufficiently large, where the upper bound was always assumed by the irreducible component of {\it totally geodesic} branched superminimal immersions and the lower bound was assumed by all {\it nontotally geodesic} irreducible components of $\mathcal{M}_{6,1}(T)$ for any torus $T$. It is shown, via deformation theory, in this note that for $d=8g+1+3k$, $k\geq 0$, and any Riemann surface $X$ of $g\geq 1$, the above lower bound is assumed by at least one irreducible component of $\mathcal{M}_{d,g}(X)$.

Maximal isotropy groups of Lie groups related to nilradicals of parabolic subalgabras

Abstract: We consider Lie groups whose Lie algebra is the nilradical of a parabolic subalgebra of a complex simple Lie algebra, endowed with left-invariant Hermitian metrics. For such Riemannian Lie groups, we describe the Lie algebras of their maximal isotropy groups.

Exotic involutions of low-dimensional spheres and the Eta-invariant

Abstract: We give a transparent description of the one-fold smooth suspension of Fintushel-Stern's exotic involution on the 4-sphere. Moreover we prove that any two involutions of the 4-sphere are stably (i.e., after one-fold suspension) smoothly conjugated if and only if the corresponding quotient spaces (real homotopy projective spaces) are stably diffeomorphic. We use the Atiyah-Patodi-Singer eta-invariant to detect smooth structures on homotopy projective spaces and prove that any homotopy projective space is detected in this way in dimensions 5 and 6.

Hardy spaces and maximal operators on real rank one semisimple Lie groups, I

Abstract: Let $G$ be a real rank one connected semisimple Lie group with finite center. As well-known the radial, heat, and Poisson maximal operators satisfy the $L^p$-norm inequalities for any $p>1$ and a weak type $L^1$ estimate. The aim of this paper is to find a subspace of $L^1(G)$ from which they are bounded into $L^1(G)$. As an analogue of the atomic Hardy space on the real line, we introduce an atomic Hardy space on $G$ and prove that these maximal operators with suitable modifications are bounded from the atomic Hardy space on $G$ to $L^1(G)$.

Decomposition of Killing vector fields on tangent sphere bundles

Abstract: Given an orientable Riemannian manifold, we consider the bundle of oriented orthonormal frames and the tangent sphere bundle over it, which admit natural Riemannian metrics defined by the Riemannian connection. We show that there is a natural homomorphism between the Lie algebras of fiber preserving Killing vector fields on these bundles. In particular, for any orientable Riemannian manifold of dimension two, we show that the homomorphism is extended to an isomorphism between these Lie algebras.

The first eigenvalues of finite Riemannian covers

Abstract: There exists a Riemannian metric on the real projective space such that the first eigenvalue coincides with that of its Riemannian universal cover, if the dimension is bigger than 2. For the proof, we deform the canonical metric on the real projective space. A similar result is obtained for lens spaces, as well as for closed Riemannian manifolds with Riemannian double covers. As a result, on a non-orientable closed manifold other than the real projective plane, there exists a Riemannian metric such that the first eigenvalue coincides with that of its Riemannian double cover.

Saturation of the approximation by spectral decompositions

Abstract: We shall give a saturation class for approximations by eigenfunction expansions of the Laplacian in an open domain in the Euclidean space.

On the second variation of the identity map of a product manifold

Abstract: . The main aim of this paper is to compute the index and the nullity of theidentity map of S n x S m and S n x T m . In order to obtain this we establish a rather generalresult on the spectrum of the Hodge-Laplacian on fc-forms on a product manifold, which couldprove useful in other contexts. 1. Introduction. If / : (M, g) -> (N, h) is a smooth map between compact orientedRiemannian manifolds, its energy is defined by (1.1) E{f) = \ f \df\ 2 dυ M . 2 JM Then a harmonic map is defined to be a smooth critical point of the functional (1.1). We referto the surveys [3], [4] for motivations and background on harmonic maps.Let f~ ι TN be the induced vector bundle by / over M and C(f~ ι TN) the space of allsections of f~ ι TN. The second variation formula for a harmonic map / was first obtainedin [8] and [9]. It is given by Hf(υ, w)= f (V f ~ lτN υ, V f ~ XτN w) - Trace (R N (df 9 v)df, w)dv M (1.2) J M= / (J f v,w)dυ M , JM for all v, w e C(f~ ι TN), where V f ~ l ™ = V is the connection on