Showing papers in "Tohoku Mathematical Journal in 1999"

On mumford's construction of degenerating abelian varieties

Abstract: For a one-dimensiona l family of abelian varieties equipped with principal theta divisors a canonical limit is constructed as a pair consisting of a reduced projective variety and a Cartier divisor on it. Properties of such pairs are established.

Quasi-einstein totally real submanifolds of the nearly kahler 6-sphere

Abstract: We investigate Lagrangian submanifolds of the nearly Kahler 6-sphere. In particular we investigate Lagrangian quasi-Einstein submanifolds of the 6-sphere. We relate this class of submanifolds to certain tubes around almost complex curves in the 6-sphere. 1. Introduction. In this paper, we investigate 3-dimensional totally real submani- folds M3 of the nearly Kahler 6-sphere S6. A submanifold M3 of S6 is called totally real if the almost complex structure / on S6 interchanges the tangent and the normal space. It has been proven by Ejiri ((El)) that such submanifolds are always minimal and orientable. In the same paper, he also classified those totally real submanifolds with constant sectional curva- ture. Note that 3-dimensional Einstein manifolds have constant sectional curvature. Here, we will investigate the totally real submanifolds of S6 for which the Ricci tensor has an eigenvalue with multiplicity at least 2. In general, a manifold Mn whose Ricci tensor has an eigenvalue of multiplicity at least n — 1 is called quasi-Einstein. The paper is organized as follows. In Section 2, we recall the basic formulas about the vector cross product on RΊ and the almost complex structure on S6. We also relate the standard Sasakian structure on S5 with the almost complex structure on S6. Then, in Section 3, we derive a necessary and sufficient condition for a totally real submanifold of S6 to be quasi-Einstein. Using this condition, we deduce from (C), see also (CDVV1) and (DV), that totally real submanifolds M with 8M = 2 are quasi-Einstein. Here, 8M is the Riemannian invariant defined by

A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension

Abstract: The recent two proofs for the (weak) factorization theorem for birational maps, one by W{\l}odarczyk and the other by Abramovich-Karu-Matsuki-W{\l}odarczyk rely on the results of Morelli. The former uses the process for $\pi$-desingularization (the most subtle part of Morelli's combinatorial algorithm), while the latter uses the strong factorization of toric (toroidal) birational maps directly. This paper provides a coherent account of Morelli's work (and its toroidal extension) clarifying some discrepancies in the original argument.

On a twisted de Rham complex

Abstract: We show that, given a projective regular function f on a smooth quasi-projective variety over C, the corresponding cohomology groups of the algebraic de Rham complex with twisted differential d-df and of the complex of algebraic forms with differential df have the same dimension (a result announced by Barannikov and Kontsevitch). We generalize the result to de Rham complexes with coefficients in a mixed Hodge Module.

Representation of flat Lagrangian $H$-umbilical submanifolds in complex Euclidean spaces

Abstract: The author proved earlier that, a Lagrangian //-umbilical submanifold in complex Euclidean «-space with n>2 is either a complex extensor, a Lagrangian pseudo-sphere, or a flat Lagrangian //-umbilical submanifold. Explicit descriptions of complex extensors and of Lagrangian pseudo-spheres are given earlier. The purpose of this article is to complete the investigation of Lagrangian //-umbilical submanifolds in complex Euclidean spaces by establishing the explicit description of flat Lagrangian //-umbilical submanifolds in complex Euclidean spaces. 1. Statements of theorems. We follow the notation and definitions given in (2). In order to establish the complete classification of Lagrangian //-umbilical submanifolds in Cn we need to introduce the notion of special Legendre curves as follows. Let z: I^S2n~ι cC"beaunit speed Legendre curve in the unit hypersphere S2n~1 (centered at the origin), i.e., z — z(s) is a unit speed curve in S2n~1 satisfying the condition: (z'(s\ iz(s)) = 0 identically. Since z = z(s) is a spherical unit speed curve, = 0 identically. Hence, z(s), iz(s% z'(s\ iz'(s) are orthonormal vector fields defined along the Legendre curve. Thus, there exist normal vector fields P 3, ...,/>" along the Legendre curve such that (1.1) Φ), ι'Φ), z'(s\ iz\s\ PM iPsis), > Pn(s), iPn(s) form an orthonormal frame field along the Legendre curve. By taking the derivatives of = 0 and of =0, we obtain =0 and = — 1, respectively. Therefore, with respect to an orthonormal frame field chosen above, z" can be expressed as

A weighted norm inequality for rough singular integrals

Abstract: We prove a weighted norm inequality for homogeneous singular integrals when only an Hι-s'ιze condition is assumed on the restriction of the kernel to the unit sphere. We also give several applications of this inequality.

Quotients of toric varieties by the action of a subtorus

Abstract: We consider the action of a subtorus of the big torus on a toric variety. The aim of the paper is to define a natural notion of a quotient for this setting and to give an explicit algorithm for the construction of this quotient from the combinatorial data corresponding to the pair consisting of the subtorus and the toric variety. Moreover, we study the relations of such quotients with good quotients. We construct a good model, i.e. a dominant toric morphism from the given toric variety to some ``maximal'' toric variety having a good quotient by the induced action of the given subtorus.

The minimal model theorem for divisors of toric varieties

Abstract: The minimal model conjecture says that if a proper variety has non-negative Kodaira dimension, then it has a minimal model with abundance and if the Kodaira dimension is — oo, then it is birationally equivalent to a variety which has a fibration with the relatively ample anti-canonical divisor. In this paper, first we prove this conjecture for a ^-regular divisor on a proper toric variety by means of successive contractions of extremal rays and flips of the ambient toric variety. Then we prove the main result: for such a divisor with the non-negative Kodaira dimension there is an algorithm to construct concretely a projective minimal model with abundance by means of \"puffing up\" the polytope. Introduction. Let k be an algebraically closed field of arbitrary characteristic. Varieties in this paper are all defined over k. Let X be a proper algebraic variety. A proper algebraic variety Y is called a minimal model of X, if (1) Y is birationally equivalent to X, (2) Y has at worst terminal singularities and (3) the canonical divisor Ky is nef. A minimal model Y is said to have abundance if the linear system \\mKγ\\ is base point free for sufficiently large m. The minimal model conjecture states: an arbitrary proper variety with K > 0 has a minimal model with abundance while an arbitrary proper variety with K = — oo has a birationally equivalent model Y with at worst terminal singularities and a fibration Y -> Z to a lower dimensional variety with — Ky relatively ample. The conjecture is classically known to hold in the 2-dimensional case. In the 3-dimensional case the conjecture for k = C is proved by Mori [4] and Kawamata [3], while it is not yet proved in higher dimension. As a special case of higher dimension, Batyrev [1] proved, among other results, the existence of a minimal model for a A -regular anti-canonical divisor of a Gorenstein Fano toric variety Tχ(A)< In this paper, first in Section 1 we prove the minimal model conjecture for every Aregular divisor X on a toric variety of arbitrary dimension by means of successive contractions of extremal rays and flips which are introduced by Reid [7]. By Bertini's theorem, for a field k of characteristic 0, the minimal model conjecture thus holds for a general member of a base point free linear system on a proper toric variety over k. An important point of this part is providing with a technical statement Corollary 1.17 which is used in the following sections. Then in Sections 2 and 3 we prove the main result: for a Z\\-regular divisor with K > 0, there exists an algorithm to construct concretely a projective minimal model with abundance by means of \"puffing up\" the polytope corresponding to the adjoint divisor. The advantage of 1991 Mathematics Subject Classification. Primary 14M25; Secondary 14Q15. Partially supported by the Grant-in-Aid for Scientific Research (No. 09640016), the Ministry of Education, Japan.

Boundary layers in a semilinear parabolic problem

Abstract: We study a singular perturbation problem for a certain type of reaction diiusion equation with a space-dependent reaction term. We compare the eeect that the presence of boundary layers versus internal layers has on the existence and stability of stationary solutions. In particular, we show that the associated eigenvalues are of diierent orders of magnitude for the two kinds of layers.

Difference spectrum and spectral synthesis

Abstract: As an aid in understanding sets of synthesis for the Fourier algebra A(G) of a locally compact abelian group G, the difference spectrum Δ{E) for a closed set E in G is studied. Numerous relations involving difference spectra of unions, intersections and cartesian products are obtained and their implications on unions, intersections and cartesian products of sets of spectral synthesis are deduced. The set Λ(E) of locally nonsynthesizable points of E is introduced and its relation with Δ{E) is discussed. The concept of ^-difference spectrum is introduced and is used to study weak spectral synthesis. Local methods are employed throughout. Introduction. The concept of a function belonging locally to an ideal at a point has been studied and exploited in spectral synthesis for a long time. If /, / are two closed ideals of A(G) (the Fourier algebra of a locally compact abelian group G), the set Δ{I, J) of points where neither / nor / is locally contained in the other has also been utilized in the study of spectral synthesis (e.g., Katznelson's proof [5, p. 36] and Stegeman's proof [8] of extensions of Helson's result and Saeki [6]) under different notations. For a closed subset E of G let Δ{E) = Δ(I{E\\ J(E)\\ where I(E) and J(E) are respectively the largest and the smallest closed ideals of A(G) with cospectrum E. Δ(E\\ the difference spectrum of E, has been studied and systematically exploited in problems on spectral synthesis by Saeki [6], Stegeman [9] and Salinger and Stegeman [7]. Using local techniques and difference spectrum some results on unions and intersections of sets of synthesis have been recently given by the authors in [2]. In this paper, besides giving some examples of difference spectrum and further results on difference spectra and spectral synthesis, we introduce the concept of ^-difference spectrum and use it to study weak synthesis. We also introduce Λ(E), the set of 'locally non synthesizable points' of E, which is closely related to Δ(E). 1. The difference spectrum. For spectral synthesis we generally follow the notations of [2], [7] and [9]. As general references for harmonic analysis, we cite [1] and [5]. We recall the definition of Δ(E\\ the difference spectrum of a closed set E: Δ{E) = {x: I(E) ΦXJ(E)} so that Δ(E) is a union of perfect subsets of dE, the boundary of E. E is a set of synthesis (an 5-set) if and only if Δ{E) = 0. No computation of Δ(E) has been given in the literature. So we start with some examples. We make use of some results of the next section. 1991 Mathematics Subject Classification. Primary 43A45; Secondary 43A46. 66 T. K. MURALEEDHARAN AND K. PARTHASARATHY EXAMPLE 1.1. (i) It is a well-known result of Laurent Schwartz that S, the unit sphere, is not of synthesis in R + , n>2. Thus Δ(S)φ0. By rotation invariance of J(S) and of I(S\\ it follows that Δ(S) = S\\ (ii) We can get an example with 0 = Δ(E)ΦE as follows. Let E be the union of S and a line segment, for instance. Then Δ(E) = S (e.g., use Lemma 2.2 below). (iii) Δ(S x S) = S x S(n, m > 2). (iv) If E is a set of synthesis in G, then Δ(SxE) = S xE. (For (iii) and (iv), use Lemma 2.7 below and (i) above.) (v) If S<=S is a spherical cap, n>2, then J(S) = S. (Use Lemma l(ii) of [2].) Reiter [5, p. 40] has proved the following local characterization of an 5-set (Wiener set in Reiter's terminology): If E is a closed set in G such that each point of E has a closed relative neighbourhood in E which is an *S-set for A(G), then E itself is an S-set. So it seems natural to introduce the following set, which also gives some measure of nonsynthesis for E. We define Λ(E) = {x e E: x has no closed relative neighbourhood in E which is an £-set} . Points of Λ(E) may be called locally nonsynthesizable points of E and, by Reiter's result, E is an S-set if and only if Λ(E) = 0. It would be natural to investigate the relation between Δ(E) and Λ(E). Here is a first step. We recall that j(E) = {feA(G):f vanishes in some neighbourhood of E) so that J(E)=](E). LEMMA 1.2. Δ(E)αΛ(E). PROOF. If xφA(E), then there is a closed relative neighbourhood V of x in E which is an S-set. Choose a neighbourhood W of x with En W^ V. Choose keA(G) such that k = 1 near x and supp&c W. Let f eI(E). Since Fis an S-set, for each positive integer n, there is a gnej(V) with \\\\f—gn\\\\<\\/n\\\\k\\\\. Then gnkej(E) and fk = \\imgnkeJ(E). Thus fexJ(E), so x COROLLARY 1.3. 7/\" £ is α closed set in G and if each point of dE has a closed relative neighbourhood in E which is an S-set, then E is an S-set. PROOF. A(E)<=Λ(E)ndE. Observe that E\\A(E) is relatively open in E, so A(E) is relatively closed in E. Since E itself is closed, this implies that A(E) is always closed (in contrast to Δ (E)). Moreover, from the definition, it does not seem to follow that A(E) c dE. (However, in the case of R this inclusion does hold.) When is Δ(E) = A(E)Ί We begin by proving the following result. THEOREM 1.4. IfE is a closed subset oflR (or ofT, the unit circle), then Δ(E) = A(E). PROOF. In view of Lemma 1.2 we have to prove that A(E)czΔ(E). Let xeA(E). DIFFERENCE SPECTRUM AND SPECTRAL SYNTHESIS 67 Let V=[x — ε, x + έ] nE be a closed relative neighbourhood of x. Then Δ(V)Φ0. If yeΔ(V) then yedE. Indeed, suppose yeΔ(V) and y lies in the interior of E. Let / be a closed interval around y with la E. Then Vn I— \\x — ε, x + ε] n / is a closed interval and is a relative neighbourhood of y in F, so yφΛ(V) and hence yφΔ(V). Now /I(F), being a union of perfect sets (actually a perfect set in this case), is uncountable and E\\V has only two possible limit points in V. Hence we can choose yeΔ(V) such that y is not a limit point of E\\V. Then there is a neighbourhood W of y with Wn(E\\V) = 0. Choose keΛ(G) such that k=\\ near 7 and suppkaW. Since yeΔ(V), there is an /e/(F) with fφyJ(V). But fkel(E), fk=y /and fφyJ{V)=>J(E). Hence >>eΛ(£). Thus, for each positive integer «, we get a j n e [ x 1/w, x + 1/n] nE with >>„eΔ(E). Since J(2s) is closed, x = \\imyneΔ(E). Essentially the same proof holds for T. REMARK 1.5. It is likely that the same result holds for R (and for T) as well, but we are unable to settle this. (A more general conjecture would be: Δ{E) = A(E)ndE when G is metrizable.) 2. Spectral synthesis. Here we discuss some results on unions, intersections and cartesian products of sets of synthesis using the difference spectrum. Some of the results given below on difference spectra have already been made use of in the previous section to compute some examples on difference spectrum. The union problem for sets of spectral synthesis is a central unsolved problem in the subject. There have been several attempts, giving rise to partial results. As our contribution in this direction, we obtain some relations between the difference spectra of two closed sets Eu E2 and their union Ex vE2. We make use of local equality of sets. Recall that E=XF means En V=Fn V for some neighbourhood V of x. LEMMA 2.1. Let E, Fbe closed sets in G and let xeG. IfE=x F, then (i) J(E) = x J(F) and (ii) Δ(E) = XΔ(F). PROOF. Suppose Fis a neighbourhood of x such that En V=Fn V. (i) Choose a neighbourhood W of x with Wa V and a keA(G) such that k— 1 near x and supp& c W. Let feJ(E)9 so /=lim/n, fnej(E). Then kfnej(F), kf=\\imkfneJ(F) and f=xkf. Thus J(E) a x J(F) and the result follows by interchanging E and F. (ii) We prove that Δ(E)n V=Δ(F)n V. Let yeΔ(E)n V. Now choose a neighbourhood W of y with Wa V and choose a keΛ(G) supported in W with k=\\ near y. Since yeΔ(E) there is an fel(E) such that fφyJ(E). But E=yF and so kf=yfφyJ(F) by (i), whereas kfeI(F). Thus yeΔ(F). This proves that Δ(E)n VαΔ(F)n Fand, by symmetry, (ii) follows. We shall also make use of the observations that the boundary of a closed set E in G is dE={xeE: 0φEφxG} and that the inclusion AαBvC holds if AczxB for each xφ C for subsets A, B, C of G. 68 T. K. MURALEEDHARAN AND K. PARTHASARATHY LEMMA 2.2. Let Eί, E2 be closed sets in G and let £ 1 2 = {xe dEx n dE2 n d(E1 u E2): x is a limit point of E1AE2} , where EίAE2 denotes the symmetric difference. Then Δ(E1υE2)<=.Δ{E1)\\)Δ{E2)υE12. PROOF. Let xφE12. Then xφdE1 or xφdE2 or xφd(Ex uE2) or x is not a limit point of EXAE2. If xφdEu either Eί=x0 or Eί=xG, that is, E1uE2=xE2 or EίuE2=xG and so J(£Ί u£ I 2) = xzl(£'2) or A(E1uE2) = x0. The cases xφδE2, xφdiE^uE^ are similar. If x is not a limit point of EίAE2, then E1ΔE2=x09 so EίϋE2=xE1 and z l ^ u£ t 2) = xzl(J£ t 1). Thus in any case xφE12 implies A(Eί \\JE2)<=X A(E1)\\JA(E2). Hence the result follows. A relation in the reverse direction is given in [2]. A consequence of Lemma 2.2 is the following result (a slight improvement of a result of Saeki [6]). THEOREM 2.3. Let Eu E2 be S-sets. If there exists a C-set C such that uE2, then E1 uE2 is an S-set. PROOF. This is immediate from Lemma 2.2 and the result of Stegeman [9] that E is an S-set if there is a C-set C with A(E)a CczE. COROLLARY 2.4. Let Eu E2 be two S-sets and suppose Ex nE2 is relatively open in Eί\\jE2, then Eγ u E2 is an S-set. PROOF. In this case El2 = 0. It is well-known that the intersection of two S-sets need not be an S-set; indeed this phenomenon occurs in any nondiscrete G (see [3]). The following lemma and its corollary, giving a sufficient condition for the intersection of two S-sets to be an

Existence and continuous dependence of mild solutions to semilinear functional differential equations in Banach spaces

Abstract: This paper is concerned with a general existence and continuous dependence of mild solutions to semilinear functional differential equations with infinite delay in Banach spaces. In particular, our results are applicable to the equations whose Co-semigroups and nonlinear operators, defined on an open set, are noncompact.

Duality for a class of minimal surfaces in {R}n+1

Abstract: Much is known about the geometry of a minimal surface in Euclidean space whose Gauss map takes values on a linear subspace of the quadric hypersurface. We consider minimal surfaces whose Gauss maps take values on rational normal curves. These are the non-degenerate minimal surfaces with smallest possible Gaussian images. We show that the geometry of such a minimal surface may be understood in terms of an auxiliary holomorphic curve on the total space of a line bundle over the Gaussian image. This is related to classical osculation duality. Natural analogues in higher dimensions of Enneper's surface, Henneberg's surface and surfaces with Platonic symmetries are described in terms of algebraic curves.

Reduction of the Hermitian-Einstein equation on Kählerian fiber bundles

Abstract: The technique of dimensional reduction of an integrable system usually requires symmetry arising from a group action. In this paper we study a situation in which a dimensional reduction can be achieved despite the absence of any such global symmetry. We consider certain holmorphic vector bundles over a Kahler manifold which is itself the total space of a fiber bundle over a Kahler manifold. We establish an equivalence between invariant solutions to the Hermitian– Einstein equations on such bundles, and general solutions to a coupled system of equations defined on holomorphic bundles over the base Kahler manifold. The latter equations are the Coupled Vortex Equations. Our results thus generalize the dimensional reduction results of Garćia–Prada, which apply when the fiber bundle is a product and the fiber is the complex projective line. 1991 Mathematics Subject Classification. Primary 58C25 ; Secondary 58A30, 53C12, 53C21, 53C55, 83C05.

The splitting and deformations of the generalized Gauss map of compact CMC surfaces

Abstract: We show that a non-conformal harmonic map from a Riemann surface into the Euclidean ^-sphere can be considered as a component of minimal surfaces in higher dimensional spheres. In the same principle, we show that the generalized Gauss map of constant mean curvature surfaces in the 3-sphere globally splits into two non-conformal harmonic maps into the 2-sphere. Using this, we obtain examples of non-trivial harmonic map deformations for compact Riemann surfaces of arbitrary positive genus. In particular, we give a lower bound for the nullity (as harmonic maps) of the generalized Gauss map of compact CMC surfaces in the 3-sphere. Furthermore, we obtain an affirmative answer to Lawson's conjecture for superconformal minimal surfaces in 4m-spheres.

The Faber-Krahn type isoperimetric inequalities for a graph

Abstract: In this paper, a graph theoretic analog to the celebrated Faber-Krahn inequality for the first eigenvalue of the Dirichlet problem of the Laplacian for a bounded domain in the Euclidean space is shown. Namely, the optimal estimate of the first eigenvalue of the Dirichlet boundary problem of the combinatorial Laplacian for a graph with boundary is given.