Showing papers in "Annals of Mathematics in 1982"

L’intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes

Abstract: L'existence de la valeur principale (2) dans le cas general (M quelconque) et l'inegalite (3) dans le cas particulier oui M est suffisamment petit sont dues a A. Calder6n [3]. La methode de perturbation utilisee par A. Calderon ne permet pas d'obtenir (3) dans le cas general que nous allons traiter. Par ailleurs notre demonstration de (3) n'utilise plus la variable complexe ni la representation conforme. Designons par 4: R -C une fonction lipschitzienne, a valeurs complexes, telle que II4'"II < 1 et appelons Kn(x, y) le noyau singulier (4(x) 4(y))n/ (x y)"'. Alors on a le resultat precis suivant.

Threefolds whose canonical bundles are not numerically effective

Abstract: Let X be an arbitrary nonsingular projective 3-fold whose canonical bundle is not numerically effective. Then we have: (i) X contains an exceptional divisor of several types, which we classify explicitly, (ii) X has a morphism to a projective nonsingular surface whose fibers are conics, (iii) X has a morphism to a projective nonsingular curve whose general fibers are Del Pezzo surfaces, or (iv) X is a Fano 3-fold with Picard number 1.

Real hypersurfaces, orders of contact, and applications

Abstract: A fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the Cauchy-Riemann equations. Suppose that D is an open domain in Cn, and that its boundary, M, is a smooth real submanifold of Cn. How does the geometry of M influence the function theory on D? One approach to this question is the a-Neumann problem, [6]. There one studies the inhomogeneous Cauchy-Riemann equations au -a where a is a (0, q) form with distribution coefficients on the closed domain D U M satisfying the necessary compatibility condition a a 0. The a-Neumann problem constructs a particular solution u which is orthogonal to the null space of the complex Laplacian. Kohn [10a] has solved this problem and has shown that good local regularity properties for u in terms of a follow from so-called subelliptic estimates. It is natural to seek geometric conditions on M that are necessary and sufficient for these estimates. In this paper we will be concerned with a certain geometric condition on M that we feel is the right one for subellipticity. Although the motivation for the questions discussed here comes from partial differential equations, the techniques come from algebraic geometry. To see why, we state theorems of Kohn, Greiner, and Catlin concerning subellipticity. Kohn [1Oa] has discovered a sufficient condition for a subelliptic estimate near a fixed point p in M. In case D is pseudoconvex and is contained in C2, this condition is equivalent to the finiteness of order of tangency of every complex analytic (one dimensional) manifold with M at p. Greiner [13] established the necessity of this condition in

Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature

Abstract: Let N be a three dimensional Riemannian manifold. Let E be a closed embedded surface in N. Then it is a question of basic interest to see whether one can deform : in its isotopy class to some "canonical" embedded surface. From the point of view of geometry, a natural "canonical" surface will be the extremal surface of some functional defined on the space of embedded surfaces. The simplest functional is the area functional. The extremal surface of the area functional is called the minimal surface. Such minimal surfaces were used extensively by Meeks-Yau [MY21 in studying group actions on three dimensional manifolds. In [MY2], the theory of minimal surfaces was used to simplify and strengthen the classical Dehn's lemma, loop theorem and the sphere theorem. In the setting there, one minimizes area among all immersed surfaces and proves that the extremal object is embedded. In this paper, we minimize area among all embedded surfaces isotopic to a fixed embedded surface. In the category of these surfaces, we prove a general existence theorem (Theorem 1). A particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area. We also prove that there exists an embedded sphere of least area enclosing a fake cell, provided the complementary volume is not a standard ball, and provided there exists no embedded one-sided RP2. By making use of the last result, and a cutting and pasting argument, we are able to settle a well-known problem in the theory of three dimensional manifolds. We prove that the covering space of any irreducible orientable three dimensional manifold is irreducible. It is possible to exploit our existence theorem to study

Characteristic Exponents and Invariant Manifolds in Hilbert Space

Abstract: The multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds (Pesin theory) are extended to differentiable dynamical systems on Hilbert manifolds under some compactness assumptions. The results apply to partial differential equations of evolution and also to non-invertible maps of compact manifolds.

Sur certains types de representations p-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti-Tate

Abstract: 1. Representations de Hodge-Tate 532 2. Construction du corps BDR 534 3. Representations de de Rham 545 4. L'anneau B 549 5. Representations cristallines et potentiellement cristallines .. 560 6. Applications aux groupes p-civisibles 564 Appendice: Quelques conjectures sur la cohomologie des varietes algebriques sur les corps locaux. 569 Bibliographie 576

The Fenchel-Nielsen deformation

Abstract: The uniformization theorem provides that a Riemann surface S of negative Euler characteristic has a metric of constant curvature -1. A hyperbolic structure can be understood in terms of its deformations. Unfortunately, the variation of the hyperbolic metric, arising from a deformation, is not determined by local data. If z is a conformal coordinate for S and A, II H II 00 < 1, a Beltrami differential then dw = dz + p dJ defines a new conformal coordinate and structure SI. The tensor p may vanish on the open set 0 C S and yet the hyperbolic metrics of S and SI do not coincide on (. This phenomenon presents a s~erious difficulty in the deformation theory of hyperbolic structures. The Fenchel-Nielsen (twist) deformation does not share this defect. The deformation is defined by cutting the surface along a simple closed geodesic, rotating one side of the cut relative to the other, and attaching the sides in their new position. A geodesic intersecting the cut is deformed to a broken geodesic. The hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface. The deformation is concentrated at the cut. The basic idea for the deformation appears in the work of Fricke-Klein, Dehn, and Fenchel-Nielsen [71, [8]. In the Fenchel-Nielsen manuscript the deformation was considered extensively; coordinates for Teichmiller space are defined in terms of the deformation. The proper definition is given by a Fuchsian group construction. Calculations are made in terms of unimodular matrices. This approach is often cumbersome. Our basic objective is to give a description of the deformation in terms of quasiconformal mappings. The Bers embedding of Teichmiiller space is central to the analytic theory of deformations. Using our characterization we calculate the first variation of the Bers embedding for the Fenchel-Nielsen deformation. The result is a Poincare series 6*, which already appears in the work of Petersson [16].

Holomorphic semi-groups and the geometry of Banach spaces

Abstract: According to a fundamental theorem of Dvoretzky (cf. [24], [25]), any infinite dimensional Banach space X contains for each integer n and each E > 0 a subspace Xn which is (1 + e)-isomorphic to 12n) In the classical Banach spaces Lp, and 1p, with 1 < p < x, it is well known that this result can be improved: These spaces contain "uniformly complemented ln)7's" (this means that we can find subspaces Xn as above with the additional property that there exist projections P.: X -* Xn such that supn 11 Pn 11 < cx). Moreover, it is also well known that this property no longer holds for 11 or lo. This motivated the question (cf. [13]) to determine which infinite dimensional Banach spaces contain "uniformly complemented 1(n)'s." We will prove below that all uniformly convex spaces have this property. Moreover, we will show that a Banach space X verifies this property "locally" (see definition 2.10 for more precision) if and only if X does not contain l(n)'s uniformly. Progressively, the work of previous authors (cf. [14], [12] and particularly [5]) has reduced such so-called "geometric" questions to a purely analytic problem consisting of verifying in X a certain inequality referred to as "K-convexity". In this paper, we connect this inequality with the holomorphy of a certain semi-group of operators on Lp(X) and we show how this leads to the results already mentioned.

The image of J in the EHP sequence

Abstract: The EHP sequence is based on a result of James [J] who showed that there is H a map H such that S _> Q Sn+- > S2n+1 is a fibration when localized at 2. The map Sn ..> Sn+ is usually labeled E. The boundary homomorphism in the homotopy sequence is usually labeled P. For our purposes it will be most convenient to combine all the EHP sequences into one system. This gives the following filtration of O201:S? = Q(S?):

The $L^2$-index theorem for homogeneous spaces of Lie groups

Abstract: The L2-index theorem for covering spaces of Atiyah [3] and Singer [20] asserts that, given a discrete group F acting smoothly and freely on a manifold M with compact quotient M = F \ M, and an elliptic differential operator D on M which is F -invariant and thus descends to an elliptic differential operator D on the compact manifold M, then the P-index indrD = dimrKerD dimrKerD* of D coincides with the ordinary index ind D = dim Ker D dim Ker D* of D. Here Ker D is the space of L2-solutions of Du = 0, and dim r denotes the dimension function corresponding to the trace (on the commutant of F acting on the Hilbert space of L2-sections over M) naturally associated to P. The importance of this theorem lies in the fact that indD > 0 implies the existence of nontrivial L2-solutions for the equation Du = 0, and as such it was used in a crucial way by Atiyah and Schmid [5] to construct explicit realizations of the discrete series representations for semisimple Lie groups. Indeed, if G is a Lie group which possesses a discrete, torsion-free, cocompact subgroup F, and H is a compact subgroup of G, then the L2-index theorem applied to the covering space M = G/H of M =F \ G/H, combined with the index formula of Atiyah-Singer [6], yields existence results for L2-solutions of G-invariant elliptic equations on the homogeneous space G/H. It is relevant to note that, with D denoting this time a G-invariant elliptic differential operator on G/H, the ratio between the F-index of D and the covolume of F gives a number independent of F. This real number, indG D, can be in fact intrinsically defined as the difference of the two "formal degrees" corresponding to the representations of G on Ker D and Ker D* respectively, and hence makes sense for any unimodular Lie group G, even when it has no discrete, cocompact subgroups (which, outside the semisimple case, is the generic

T, T-1 transformation is not loosely Bernoulli*

Abstract: The T, T-' problem, open since 1971, is to demonstrate that the T, T` transformation, a naturally arising K transformation, is not Bernoulli. It is shown here not even to be loosely Bernoulli.

Strongly pseudoconvex CR structures over small balls* Part II. A regularity theorem

Abstract: The purpose of the present paper is to prove the regularity theorem for the solution of the generalized Neumann boundary value problem for a strongly pseudoconvex CR structure, say ?T", over a small ball. This will be based on the a priori estimate established in [I]. Namely, we take an admissible distance function t to a reference point (cf. (2.12) in [I]) and consider a sufficiently small ball Ur: t < r. Denote by D the differential operator in the complex associated with 0T" and consider the metric given by a Levi-form. Set b Dt , which is a function of U, We consider the Hilbert space of L2 norm with weight factor b2a, 11 11 a and (, )a denoting its norm and inner product, respectively. Let aD* be the adjoint of D. Set Qa(U v) =(Du, Dv)a + (aD*u'aD*v)a for 0T" q-forms u, v. The generalized Neumann boundary value problem is to solve the following equation: For a given 0T" q-form a and an unknown 0T" q-form u withuLDt= 0,

Regular tessellations of surfaces and (p, q, 2)-triangle groups

Abstract: The purpose of this paper is a study of the existence and classification of certain regular tessellations of surfaces and their relationship with the Schwarz triangle groups of type (p, q, 2). These tessellations are the natural generalizations to arbitrary surfaces of the spherical tessellations corresponding to the five Platonic Solids. A regular tessellation of type {p, q}, a {p, q}-pattern, for short, on a surface M is a tessellation of M by p-sided faces such that the valence of each vertex is q. (A more precise definition appears in ? 1.) No global symmetry is assumed. A { p, q}-pattern on M is said to be geometric if M admits a Riemannian metric of constant curvature with respect to which the edges of the pattern are geodesic arcs of equal length and the interior angles at all vertices in all faces are (2 7/q). If the curvature happens to he zero then we further insist that the area of each face be 1. It is natural to call two {p, q}-patterns on M

Strongly Pseudoconvex CR Structures over Small Balls Part III. An Embedding Theorem

Abstract: The purpose of the present paper is to prove that any germ of strongly pseudoconvex CR structure (on a manifold of real dimension ? 9) is realized by an embedding in the complex euclidean space of real codimension 1. It is well known that the above is true as formal power series. Hence we find an embedding such that the induced structure is very close to the given one on a small neighborhood of the reference point. We correct our embedding by Newton's method. We then apply the Moser-Nash procedure (cf. [4])' to prove that the sequence of embeddings obtained by the successive correction converges on a sufficiently small neighborhood of the reference point to an embedding which realizes the given structure. To explain how the correction is made, let D be the differential operator in the complex associated with the given structure. When f = (f..... ,f) is an approximating embedding defined on U, we set Ox = Dfx. If we can find gx such that (*) DgA = ON and f g is still an embedding, f is completely corrected. Unfortunately, we do not know how to find such a g.. However, the Neumann boundary value problem in Parts I and II for the differential operator Df (which is associated with the induced structure) is a tool to handle such question. By Newton's method we mean that we make the best correction using Df instead of D. This is done as follows: OA is not a differential form in the complex of Df. Hence we replace (01,. OJ) by its projection (O',... , On) to the complex of Df. Note that 0A may no longer be in the kernel of Df and our domain may not be distinguished. We first construct a distinguished distance function tf on U and consider a ball U= {tf ? r}

Strongly pseudoconvex CR structures over small balls. Part I. An a priori estimate

Abstract: The purpose of the present series of papers is to develop the theory of harmonic integrals on strongly pseudoconvex CR structures over small balls along the line developed by D. C. Spencer, C. B. Morrey, J. J. Kohn and L. Nirenberg ([5], [6], [7], [8]) for strongly pseudoconvex complex manifolds. We consider a strongly pseudoconvex CR structure on a manifold of real dimension 2n-1. In Part I we establish the a priori estimate for the Neumann boundary value problem on the complex associated with the structure, in the case the structure is induced by an embedding in C" and restricted to a small ball of special type, provided I c q c n 3 where q is the degree of differential forms we consider. In Part II we derive the regularity theorem of solutions of the Neumann boundary value problem based on the a priori estimate of Part I. As an application we prove in Part Il that, when n ? 5, the structure is realized on a neighborhood of a reference point by an embedding in Cn. We note here that L. Nirenberg gave an example of a strongly pseudoconvex CR structure which cannot be induced by such an embedding when n =2, [9]. L. Boutet de Monvel proved that, in the case of such structures over a compact manifold, we can find such an embedding when n -3, [1]. There are two major differences between the cases of strongly pseudoconvex integrable almost complex structures and CR structures. Firstly the principal parts of the associated Laplacians in CR structures are no longer elliptic. Secondly, we have characteristic points on the boundary. As a result, when we follow over the predecessor's calculation in our case, we end up with terms which may not be absorbed in the main non-negative terms. Fortunately these leftover terms depend mostly on the Levi-form of the CR structure and the hessian form (with respect to the CR structure) of the equation of the boundary of our

Algebraic cycles and vector bundles over affine three-folds

Abstract: On any smooth affine surface over an algebraically closed field, it is easy to construct rank two projective modules with given determinant as the first Chern class and an arbitrary rational equivalence class of a zero cycle as the second Chern class. We will briefly indicate this method in Section 1. Here we construct projective modules over smooth affine 3-folds with prescribed Chern classes and of the right rank. That is to say, we construct rank two projective modules with prescribed first and second Chem classes and rank three projective modules with prescribed first, second and third Chern classes (Theorem 2.1). As a corollary, using Roitman's theorem on torsion in zero cycles and Suslin's cancellation theorem, one gets that a rank three projective module over a smooth affine threefold with top Chem class zero has a free direct summand of rank one (Cor. 2.4). For example, over a rational threefold any projective module splits into a free module and a rank two projective module. Thus we obtain necessary and sufficient conditions in terms of appropriate Chern classes for modules to be efficiently generated. For example, on an affine rational 3-fold any line bundle is 3-generated and any rank two projective module is 4-generated. We also prove that any maximal ideal over a smooth affine variety of dimension d is the zero of a section of a rank d projective module (Theorem 3.1). For an affine 3-fold, if A3(X) = 0, we prove the validity of the Eisenbud-Evans estimate for finitely generated modules. We do not know the answers for most of these questions in higher dimensions. Sometimes we assume that the characteristic of the ground field is different from 2, 3 and 5 due to technical reasons.**