Showing papers in "Manuscripta Mathematica in 2000"
TL;DR: In this paper, local minimizers of elliptic variational integrals with integrand f of nearly linear growth were considered, and it was shown that u has Holder continuous first derivatives in the interior of the domain Ω.
Abstract: We consider local minimizers of elliptic variational integrals with integrand f of nearly linear growth. In the scalar case N= 1 a side condition of the type u≥Φ may be incorporated, for N > 1 u is an unconstrained minimizer and f is required just to depend on the modulus of Du. We show in both cases that u has Holder continuous first derivatives in the interior of the domain Ω.
111 citations
TL;DR: In this article, the existence of an infinite number of solutions for non-homogeneous boundary value problems is established. But the results are more satisfactory than those obtained by the standard "Perturbation from Symmetry" method that was developed in various forms by Bahri-Berestycki, Struwe and Rabinowitz.
Abstract: We use a method recently devised by Bolle to establish the existence of an infinite number of solutions for various non-homogeneous boundary value problems. In particular, we consider second order systems, Hamiltonian systems as well as semi-linear partial differential equations. The non-homogeneity can originate in the equation but also from the boundary conditions. The results are more satisfactory than those obtained by the standard “Perturbation from Symmetry” method that was developed – in various forms – in the early eighties by Bahri–Berestycki, Struwe and Rabinowitz.
87 citations
TL;DR: In this article, it was shown that critical points of the total scalar curvature functional on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics.
Abstract: It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8.
54 citations
TL;DR: The authors showed that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural Higgs bundle structure which is invariant under the ℂ* action studied in [20].
Abstract: Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over ℚ carries a natural Higgs bundle structure which is invariant under the ℂ* action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov–Witten potential of the underlying family of Calabi–Yau threefolds.
47 citations
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TL;DR: In this article, it was shown that certain subcategories of, consisting of complete modules having a quasi-Verma flag with respect to a Levi subalgebra, admit a combinatorial description similar to Soergel's results on category.
Abstract: Abstract: We prove that certain subcategories of , consisting of complete modules having a quasi-Verma flag with respect to a Levi subalgebra, admit a combinatorial description similar to Soergel's results on category . Using the Enright completion functor we also reprove Soergel's character formula for tilting modules in and Ringel self-duality for the principal block of .
39 citations
TL;DR: For an algebraic number field k and a prime number p (if p=2, we assume that μ4⊂k), the maximal rank of a free pro-p extension of k was studied in this paper.
Abstract: For an algebraic number field k and a prime number p (if p=2, we assume that μ4⊂k), we study the maximal rank ρ
p
of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of these conjectures and we apply them to show that, essentially, ρ
k
=r
2(k)+1 if and only if k is a so-called p-rational field.
33 citations
AT&T1
TL;DR: In this paper, an infinite-dimensional inner metric Alexandrov space of nonnegative curvature was constructed, which has in one point a tangent cone whose metric is not an inner metric.
Abstract: The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric.
31 citations
TL;DR: In this paper, the authors characterized the spacelike hyperplanes in the Lorentz-Minkowski space L n + 1 as the only complete complete hypersurfaces with constant mean curvature which are bounded between two parallel hyperbolic hyperplanes.
Abstract: In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L n +1 as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L n +1 which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L n +1 which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it.
31 citations
TL;DR: For a pair of anisotropic (2n-1)-dimensional quadrics X and Y, it was shown in this paper that existence of a rational morphism Y→X is equivalent to existence of rational morphisms X→X.
Abstract: Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By Xφ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of Xφ and Xψ coincide but \(\). For a pair of anisotropic (2n-1)-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X.
30 citations
TL;DR: For the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds.
Abstract: We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds.
TL;DR: In this paper, it was shown that a non-zero point P ∈ E(k) lies in lE(k), if and only if P lies in LE (k) for almost all finite primes of k. The conditions on l under which analogous results hold for Abelian varieties were given.
Abstract: Given a prime l and an elliptic curve E defined over a number field k, we show that a non-zero point P∈E(k) lies in lE(k) if and only if P lies in lE(k)(mod ?) for almost all finite primes ? of k. We give conditions on l under which analogous results hold for Abelian varieties and with one point replaced by a finite number of points. We also construct examples to show that these conditions are essential.
TL;DR: In this article, the authors developed a gluing procedure for any k ≥ 2 and any n ≥ 3 complete immersed minimal hypersurfaces of ℝn+1 which have k planar ends.
Abstract: Paralleling what has been done for minimal surfaces in ℝ3, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝn+1 which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature.
TL;DR: In this paper, the Cartier type p-descent theorem for log smooth log-smooth modules over a base of positive characteristic p has been established, and it has been used to give an alternate proof of a result of Tsuji on closed forms fixed by Cartier operator.
Abstract: After discussing gradings by sheaves of degrees, we associate to any log scheme a canonical invertible sheaf endowed with a certain multiplicative structure, which we call its associated graded algebra. In the relative case we construct a canonical connection on this algebra. In the log smooth case over a base of positive characteristic p, we study integrable and p-integrable graded modules over this algebra, and establish a Cartier type p-descent theorem, generalizing previous results of Ogus. We apply it to give an alternate proof of a result of Tsuji on closed forms fixed by the Cartier operator
TL;DR: In this article, it was shown that locally uniform limit sc:= limc→csa exists as holomorphic map sc:D→E and has values in the boundary of a bounded symmetric domain realized as the open unit ball of a complex Banach space.
Abstract: Let D be a bounded symmetric domain realized as the open unit ball of a complex Banach space E and denote for every \(\) by sa the symmetry of D about a. We show that for every boundary point \(\) the locally uniform limit sc:= limc→csa exists as holomorphic map sc:D→E and has values in the boundary \(\) of D.
TL;DR: In this article, the authors studied transitivity conditions on the norm of JB-triples, C-algebras, JB algebra, and their preduals.
Abstract: We study transitivity conditions on the norm of JB -triples, C-algebras, JB - algebras, and their preduals. We show that, for the predual X of a JBW -triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB -triple with transitive norm is a Hilbert space. We extend to arbitrary C-algebras previously known characterizations of transitivity (20) and convex transitivity (36) of the norm on commutative C-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB -algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB -algebra with almost transitive norm is isomorphic to the field of real numbers.
TL;DR: In this article, the authors investigated epimorphisms in the category of reduced partially ordered rings (porings) and showed that the set of isomorphism classes of a given poring has a largest element (an epimomorphic hull).
Abstract: The paper contributes to the investigation of epimorphisms in the category of reduced partially ordered rings (porings) Two main questions are considered: 1) Does the set of isomorphism classes of a given poring have a largest element (an epimorphic hull)? 2) Given an epimorphic extension, or even a Prufer extension, f:A→B of porings: how closely are A and B related to each other?
TL;DR: In this article, the fundamental results of genus theory for finite (non necessary Galois) extensions of global fields were established by using narrow S-class groups, when S is an arbitrary finite set of places.
Abstract: We establish the fundamental results of genus theory for finite (non necessary Galois) extensions of global fields by using narrow S-class groups, when S is an arbitrary finite set of places. This exposition, which involves both the number fields and the functions fields cases, generalizes most classical results on this subject.
TL;DR: In this paper, the Harnack inequality for nonnegative solutions of the quasilinear equation under very general structural assumptions satisfied by functions A and B was proved for both functions.
Abstract: In this note we prove the Harnack inequality for non negative solutions of the quasilinear equation under very general structural assumptions satisfied by functions A and B.
TL;DR: In this paper, the authors describe the Jacquet-Langlands correspondence between the discrete series of G and that of G′ in terms of Carayol's parametrization of these discrete series.
Abstract: Let F be a non-Archimedean locally compact field, and let p be its residual characteristic. Put G=GLp(F) and let G′=D×, where $D$ is a division algebra with centre F and of degree p2 over F. The Jacquet–Langlands correspondence is a bijection between the discrete series of G and that of G′. We describe this explicitly, in terms of Carayol's parametrization of these discrete series.
TL;DR: In this article, a theory of finite determinacy and unfolding in the spirit of Thom-Mather theory for C∞ function germs in a finitely generated and closed ideal with three additionnal properties is proposed.
Abstract: In this paper we propose to develop a theory of finite determinacy and unfolding in the spirit of Thom–Mather theory for C∞ function germs in a finitely generated and closed ideal with three additionnal properties. It may be used as the beginning of a more systematic study of non isolated real singularities.
TL;DR: In this paper, it was shown that the links at infinity of some hypersurfaces diffeomorphic to affine spaces are knotted spheres and that the property of being M-tame depends on the algebraic coordinate system of the hypersurface when n ≥ 4.
Abstract: The aim of this paper is to continue the study started in [10] and to give a topological description of the Milnor fibre at infinity. As an application, we show that the links at infinity of some hypersurfaces diffeomorphic to affine spaces C
k
, given in [3], are knotted spheres. In the last Section of this paper we give examples which show that the property of being M-tame depends on the algebraic coordinate system of C
n
, when n≥ 4.
TL;DR: In this article, it was shown that the cohomology vanishing of the first term in the p-filtration of the sheaf of differential operators on the flag variety is a necessary condition for the variety to be D-affine.
Abstract: The flag varieties in characteristic 0 are well-known to be D-affine. In positive characteristic, however, only those in type A
1 and A
2 have been proved to be so. In this paper we will show in type B
2 the cohomology vanishing of the first term in the p-filtration of the sheaf of differential operators on the flag variety. This is a necessary condition for the variety to be D-affine.
Journal Article•
TL;DR: In this paper, the authors considered the matrix for the Satake isomorphism with respect to natural bases and gave a simple proof in the case of Chevalley groups that the matrix coefficients which are not obviously zero are in fact positive numbers.
Abstract: Abstract: We consider the matrix for the Satake isomorphism with respect to natural bases. We give a simple proof in the case of Chevalley groups that the matrix coefficients which are not obviously zero are in fact positive numbers. We also relate the matrix coefficients to Kazhdan–Lusztig polynomials and to Bernstein functions.
TL;DR: In this paper, a new solution operator for piecewise smooth q-convex intersections is proposed and Lp estimates are obtained for the solution operators of closed forms on such domains.
Abstract: We construct a new solution operator for \(\) on certain piecewise smooth q-convex intersections. Lp estimates are obtained for the solution operators of \(\)-closed forms on such domains.
TL;DR: In this paper, the authors prove the existence of a generalized moment version of the A.C.L.T, with a speed of convergence, using the notion of quasi-orthogonal systems and a Gaussian randomization technic.
Abstract: In a recent work, we indicated another formulation of the Almost Sure Central Limit Theorem (A.S.C.L.T.), with series in place of averages, by showing that the property of the A.S.C.L.T. directly follows from the theory of orthogonal sums. For, we used the notion of quasi-orthogonal systems introduced earlier by R. Bellmann, and later developed by Kac–Salem–Zygmund. The main object of this paper is to prove a similar result for irrational rotations of the torus. We prove the existence of a generalized moment version of the A.S.C.L.T., with a speed of convergence. In our strategy, we use again the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context. The proof avoid notably the use of Volny's result on the existence of good Gaussian approximations in aperiodic dynamical systems, and should also permit to be able to treat problems of comparable nature, in particular in non-ergodic cases.
TL;DR: In this paper, the Euler-Poincare characteristic χ(X,F) was derived in terms of easy local and global numerical invariants, much like the formula of Grothendieck-Ogg-Shafarevich in the case of different characteristic.
Abstract: Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible etale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π* F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincare characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves.
TL;DR: In this article, the authors studied the topological structure and homeomorphism problem of closed 3-manifolds obtained by pairwise identifications of faces in the boundary of polyhedral 3-balls.
Abstract: We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links.
TL;DR: In this article, the authors extend Okubo's result to composition algebras satisfying (x, x, x) = 0 over a field of characteristic not 2 which contains at least four elements.
Abstract: Okubo showed that a power associative composition algebra over a field of characteristic not 2 or 3 is Hurwitz. In this paper we extend Okubo’s result to composition algebras satisfying (x, x, x) = (x, x, x 2) = 0 over a field of characteristic not 2 which contains at least four elements.
TL;DR: In this article, the moduli space of stable parabolic vector bundles on a smooth complex projective surface was shown to be a non-singular quasi-projective variety naturally endowed with a family of holomorphic Poisson structures parametrized by the global sections of ω X −1(−D).
Abstract: Let X be a smooth complex projective surface and D an effective divisor on X such that H0(X,ω X −1(−D)) ≠ 0. Let us denote by ?ℬ the moduli space of stable parabolic vector bundles on X with parabolic structure over the divisor D (with fixed weights and Hilbert polynomials). We prove that the moduli space ?ℬ is a non-singular quasi-projective variety naturally endowed with a family of holomorphic Poisson structures parametrized by the global sections of ω X −1(−D). This result is the natural generalization to the moduli spaces of parabolic vector bundles of the results obtained in [B2] for the moduli spaces of stable sheaves on a Poisson surface. We also give, in some special cases, a detailed description of the symplectic leaf foliation of the Poisson manifold ?ℬ.