Showing papers in "Manuscripta Mathematica in 2003"

Multiplicity of positive solutions of a nonlinear Schrödinger equation

Abstract: We consider the multiple existence of positive solutions of the following nonlinear Schrodinger equation: where ${{p\in (1, {{N+2}\over{ N-2}})}}$ if N≥3 and p(1, ∞) if N=1,2, and a(x), b(x) are continuous functions. We assume that a(x) is nonnegative and has a potential well Ω := int a −1(0) consisting of k components ${{\Omega_1, \ldots, \Omega_k}}$ and the first eigenvalues of −Δ+b(x) on Ω j under Dirichlet boundary condition are positive for all ${{j=1,2,\ldots,k}}$ . Under these conditions we show that (PM λ) has at least 2 k −1 positive solutions for large λ. More precisely we show that for any given non-empty subset ${{J\subset\{1,2,\ldots k\}}}$ , (P λ ) has a positive solutions u λ (x) for large λ. In addition for any sequence λ n →∞ we can extract a subsequence λ n i along which u λni converges strongly in H 1 (R N ). Moreover the limit function u(x)=lim i→∞ u λni satisfies (i) For jJ the restriction u| Ω j of u(x) to Ω j is a least energy solution of −Δv+b(x)v=v p in Ω j and v=0 on ∂Ω j . (ii) u(x)=0 for ${{x\in {\bf R}^N\setminus(\bigcup_{{j\in J}} \Omega_j)}}$ .

On Fourier coefficients of automorphic forms of symplectic groups

Abstract: In this paper we study certain properties of Fourier coefficients of cuspidal representations on symplectic groups. We prove that every cuspidal representation has a nontrivial Fourier coefficient with respect to a certain type of unipotent class.

Graded rings of Hermitian modular forms of degree 2

Abstract: At first it is shown that any Siegel modular form of degree 2 and even weight can be lifted to a Hermitian modular form of degree 2 over any imaginary-quadratic number field K. In the cases \(\) and \(\) we describe the graded rings of Hermitian modular forms with respect to all abelian characters. Generators are constructed as Maaslifts or as Borcherds products. The description allows a characterization in terms of generators and relations. Moreover we show that the graded rings are generated by theta constants by analyzing the associated five-dimensional representation of PSp2(ℤ/3ℤ). As an application the fields of Hermitian modular functions are determined.

Monads on projective spaces

Abstract: Let ℰ be a vector bundle on Pn. There is a strong relationship between ℰ and its intermediate cohomology modules. In the case where ℰ has low rank, we exploit this relationship to provide various splitting criteria for ℰ. In particular, we give a splitting criterion for ℰ in terms of the vanishing of certain intermediate cohomology modules. We also show that the Horrocks-Mumford bundle is the only non-split rank two bundle on P4 with a Buchsbaum second cohomology module.

Surjectivity Criteria for p-adic Representations, Part I

Abstract: We prove general surjectivity criteria for p-adic representations. In particular, we classify all adjoint and simply connected group schemes G over the Witt ring W(k) of a finite field k such that the reduction epimorphism \(G(W_2(k))\twoheadrightarrow G(k)\) has a section.

Some local-global applications of Kummer theory

Abstract: We consider some problems in number theory which turn out to depend on various aspects of Kummer theory; among them are (1) does the assertion ``b is in the subgroup generated by a'' obey a local-global principle for points of an algebraic group over a number field; (2) if two abelian varieties have the same n-division fields for n≥1, what relation is there between them?

Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature

Abstract: Let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface in the (m+1)-dimensional unit sphere Sm+1 without umbilics. Four basic invariants of x under the Mobius transformation group in Sm+1 are a Riemannian metric g called Mobius metric, a 1-form Φ called Mobius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Mobius second fundamental form. In this paper, we prove the following classification theorem: let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface, which satisfies (i) Φ≡0, (ii) A+λg+μB≡0 for some functions λ and μ, then λ and μ must be constant, and x is Mobius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in Sm+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space Rm+1; or (iii) the image of the standard conformal map of a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional hyperbolic space Hm+1. This result shows that one can use Mobius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in Sm+1, Rm+1 and Hm+1.

Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric

Abstract: Given an orientable hypersurface M of a Lie group 𝔾 with a bi-invariant metric we consider the map N : M → 𝕊 n that translates the normal vector field of M to the identity, which is a natural extension of the usual Gauss map of hypersurfaces in Euclidean spaces; it is proved that the Laplacian of N satisfies a formula similar to that satisfied by the usual Gauss map. One may then conclude that M has constant mean curvature (cmc) if and only if N is harmonic; some other aplications to cmc hypersurfaces of 𝔾 are also obtained.

A Blow-up Theorem for regular hypersurfaces on nilpotent groups

Abstract: We obtain an intrinsic Blow-up Theorem for regular hypersurfaces on graded nilpotent groups. This procedure allows us to represent explicitly the Riemannian surface measure in terms of the spherical Hausdorff measure with respect to an intrinsic distance of the group, namely homogeneous distance. We apply this result to get a version of the Riemannian coarea forumula on sub-Riemannian groups, that can be expressed in terms of arbitrary homogeneous distances. We introduce the natural class of horizontal isometries in sub-Riemannian groups, giving examples of rotational invariant homogeneous distances and rotational groups, where the coarea formula takes a simpler form. By means of the same Blow-up Theorem we obtain an optimal estimate for the Hausdorff dimension of the characteristic set relative to C1,1 hypersurfaces in 2-step groups and we prove that it has finite Q–2 Hausdorff measure, where Q is the homogeneous dimension of the group.

Some stable vector bundles with reducible theta divisor

Abstract: Let C be a curve of genus g and L a line bundle of degree 2g on C. Let ML be the kernel of the evaluation map \(\). We show that when L is general enough, the rank g bundle ML and its exterior powers are stable, but admit a reducible theta divisor.

On endomorphisms of projective bundles

Abstract: Let X be a projective bundle. We prove that X admits an endomorphism of degree >1 and commuting with the projection to the base, if and only if X trivializes after a finite covering. When X is the projectivization of a vector bundle E of rank 2, we prove that it has an endomorphism of degree >1 on a general fiber only if E splits after a finite base change.

Lifting of morphisms to quotient presentations

Abstract: In this article we investigate algebraic morphisms between toric varieties. Given presentations of toric varieties as quotients we are interested in the question when a morphism admits a lifting to these quotient presentations. We show that this can be completely answered in terms of invariant divisors. As an application we prove that two toric varieties, which are isomorphic as abstract algebraic varieties, are even isomorphic as toric varieties. This generalizes a well-known result of Demushkin on affine toric varieties.

On the generalized critical values of a polynomial mapping

Abstract: Let be a polynomial dominant mapping and let deg f i ≤d. We prove that the set K(f) of generalized critical values of f is contained in the algebraic hypersurface of degree at most D=(d+s(m−1)(d−1)) n , where . This implies in particular that the set B(f) of bifurcations points of f is contained in the hypersurface of degree at most D=(d+s(m−1)(d−1)) n . We give also an algorithm to compute the set K(f) effectively.

Parties semi-algébriques d'une variété algébrique p -adique

Abstract: Let k be a field complete with respect to an ultrametric absolute value, let A be a finitely generated k-algebra and let X be its spectrum. We denote by X an the analytification a la Berkovich of the algebraic variety X. We say that a subset of X an is semi-algebraic if it can be defined by a boolean combination of inequalities |f|⋈λ|g| where f and g are in A, where ⋈ is a symbol belonging to { ,≤,≥} and where λ is a positive real number. In this text we show that the image of any semi-algebraic subset under an algebraic map between affine varieties is semi-algebraic, and that a semi-algebraic subset has only finitely many connected components, each of which is semi-algebraic. In fact our results concern not only algebraic varieties, but also analytic families of algebraic varieties which are parametrized by an affinoid space.

ℱK-convex functions on metric spaces

Abstract: By an ℱK-convex function on a length metric space, we mean one that satisfies fn ≥ −Kf on all unitspeed geodesics. We show that natural ℱK-convex (-concave) functions occur in abundance on metric spaces of curvature bounded above (below) by K in the sense of Alexandrov. We prove Lipschitz extension and approximation theorems for ℱK-convex functions on CAT(K) spaces.

Submanifolds with constant Möbius scalar curvature in Sn

Abstract: Let Mm be a m-dimensional submanifold in the n-dimensional unit sphere Sn without umbilic point. Two basic invariants of Mm under the Mobius transformation group of Sn are a 1-form Φ called Mobius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let Mm be a m-dimensional (m≥3) submanifold with vanishing Mobius form and with constant Mobius scalar curvature R in Sn, denote the trace-free Blaschke tensor by \(\). If \(\), then either ||A||≡0 and Mm is Mobius equivalent to a minimal submanifold with constant scalar curvature in Sn; or \(\) and Mm is Mobius equivalent to \(\) in \(\) for some c≥0 and \(\).

Unitary units and skew elements in group algebras

Abstract: Let FG be the group algebra of a group G over a field F and let * denote the canonical involution of FG induced by the map g→g −1 ,gG. Let Un(FG)={uFG|uu * =1} be the group of unitary units of FG. In case char F=0, we classify the torsion groups G for which Un(FG) satisfies a group identity not vanishing on 2-elements. Along the way we actually prove that, in characteristic 0, the unitary group Un(FG) does not contain a free group of rank 2 if FG − , the Lie algebra of skew elements of FG, is Lie nilpotent. Motivated by this connection we characterize most groups G for which FG − is Lie nilpotent and char F≠2.

Real quadratic fields with class number divisible by 3

Abstract: We shall show that the number of real quadratic fields whose absolute discriminant is ≤ x and whose class number is divisible by 3 is improving the existing best known bound of K Chakraborty and R Murty

Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals

Abstract: If u is a minimizer of ∫ΩF(|∇u|)dx−∫Ωudμ, then the pointwise estimate $$$$ can be reached. This results is obtained for a Young function F with the global Δ2 & ∇2 property. Links to applications to real analysis are given.

Lineally convex domains of finite type: holomorphic support functions

Abstract: Smooth bounded lineally convex domains of finite type constitute a natural class of domains in complex analysis, since they are locally biholomorphically invariant. A smooth family of holomorphic support functions is constructed by an almost explicit formula on every such domain. It satisfies the best possible estimates near the point of support on every two-dimensional transverse affine intersection with the domain. Together with a suitable pseudometric on these domains, it will allow to do precise quantitative complex analysis by integral kernels on them.

Twisted stability and Fourier-Mukai transform II

Abstract: In this note, we define the twisted stability for a purely 1-dimensional sheaf and study the problem of the preservation of the stability condition under the relative Fourier-Mukai transform on an elliptic surface. As an application, we compute the Hodge polynomials of some moduli spaces of sheaves on an elliptic surface. We also construct the moduli space of twisted semi-stable sheaves.

Non-standard tight closure for affine ℂ-algebras

Abstract: In this paper, non-standard tight closure is proposed as an alternative for classical tight closure on finitely generated algebras over ℂ It has the advantage that it admits a functional definition, similar to the characteristic p definition of tight closure, where instead of the characteristic p Frobenius, its ultraproduct, the non-standard Frobenius, is used This new closure operation cl(⋅) has the same properties as classical tight closure, to wit, (1) if A is regular, then 𝔞=cl(𝔞); (2) if A⊂B is an integral extension of domains, then cl(𝔞 B)∩A⊂cl(𝔞); (3) if A is local and is a system of parameters, then (Colon-Capturing); (4) if 𝔞 is generated by m elements, then cl𝔞 contains the integral closure of 𝔞 m and is contained in the integral closure of 𝔞 (Briancon-Skoda)

Centrally symmetric generators in toric Fano varieties

Abstract: We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has ``many'' pairs of centrally symmetric vertices.

Non-existence of higher-dimensional pseudoholomorphic submanifolds

Abstract: In this paper we prove that only pseudoholomorphic curves appear as J-invariant submanifolds of generic almost complex manifolds (M,J). We also prove there exist no non-trivial automorphisms or submersions of such manifolds. On the other hand we show that abundance of 1-jets of PH-submanifolds, automorphisms or submersions implies integrability of the almost complex structure.

Congruence of minimal surfaces and higher fundamental forms

Abstract: We show that minimal surfaces in space forms are determined, up to ambient isometries, by the induced metric and certain invariants which are defined in terms of the higher fundamental forms and the complex structure.

Two dimensional variational problems with linear growth

Abstract: Suppose that f: ℝnN→ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|2) if N>1. If f satisfies an ellipticity condition of the form $$$$ then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem $$$$ provided that the given boundary data u0W11(ω;ℝN) are additionally assumed to be of class L∞(ω;ℝN). Moreover, if μ<3, then the boundedness of u0 yields local C1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem $$$$

Homogeneous spaces with polar isotropy

Abstract: We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation.

Exemples de Lattès et domaines faiblement sphériques de ℂ n

Abstract: We describe the attracting basins of the origin in ℂ k+1 for the polynomial lifts of Lattes examples. We show that the boundary of these bounded pseudoconvex domains is a quotient of a compact spherical hypersurface, and we describe the singularities that appear. These domains are surprising, because they are very close to the ball, and admit non injective proper holomorphic self-maps. We also explicit some Lattes examples in dimension 2.

A note on radial graphs with constant mean curvature

Abstract: Let Ω be a smooth domain on the unit sphere 𝕊n whose closure is contained in an open hemisphere and denote by ℋ the mean curvature of ∂Ω as a submanifold of Ω with respect to the inward unit normal. It is proved that for each real number H that satisfies inf ℋ > − H ≥ 0, there exists a unique radial graph on Ω bounded by ∂Ω with constant mean curvature H. The orientation on the graph is based on the normal that points on the opposite side as the radius vector.

Perversité et variation

Abstract: Let S be the spectrum of a strictly henselian discrete valuation ring with residue characteristic p and Λ=ℤ/lνℤ, where l is a prime number ≠p and ν is an integer ≥1. For a scheme X of finite type over S and smooth over S along the special fiber Xs outside a closed point x, we study the vanishing cycles complex RΦ(Λ) and the tame variation \({{\mathop{{\rm{ Var}}}(\sigma) : R\Phi_{{{{\rm{ t}}}}}(\Lambda)_x \rightarrow R\Gamma_{{\{x\}}}(X_s,R\Psi_{{{{\rm{ t}}}}}(\Lambda))}}\), for Σ in the tame inertia group It. In particular, we show that if X is regular, flat over S of relative dimension n≥1, and Σ is a topological generator of It, then RqΦ(Λ)x=0 for q≠n and \({{\mathop{{\rm{ Var}}}(\sigma) : R^n\Phi_{{{{\rm{ t}}}}}(\Lambda)_x \rightarrow H^n_{{\{x\}}}(X_s,R\Psi_{{{{\rm{ t}}}}}(\Lambda))}}\) is an isomorphism.