Showing papers in "Annals of Mathematics in 1983"

Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

Abstract: A maximizing function, f, is shown to exist for the HLS inequality on R': 11 IXI - * fIq < Np f A , Iif IIwith Nbeing the sharp constant and i/p + X/n = 1 + 1/q, 1

Asymptotics for a class of non-linear evolution equations, with applications to geometric problems

Abstract: Soit Σ une variete de Riemann compacte et soit une fonction reguliere u=u(x,t), (x,t)∈ΣX(0,T) (T>0) satisfaisant une equation d'evolution soit de la forme ci-#7B-M(u)=f soit de la forme −u+u˙-#7B-M(u)-#7B-R(u)=f. (#7B-M(u) est l'operateur d'Euler-Lagrange d'ordre 2). On etudie le comportement asymptotique des solutions de ces equations

The Nielsen Realization Problem

Abstract: Closed, oriented surfaces of genus g > 2 admit many hyperbolic (constant Gaussian curvature -1) metrics in contrast to Mostow's rigidity theorems in higher dimensions. Only special hyperbolic surfaces have non-trivial groups of isometries, but many different, non-isomorphic groups arise for different symmetric metrics. The group of isometries of a closed hyperbolic surface M2 is always finite and the only isometry isotopic to the identity is the identity itself. As a result, hyperbolic surfaces with non-trivial groups of isometries have been a primary source for the construction of finite subgroups of the group of isotopy classes of diffeomorphisms of M2, ?TDiff(M2). An old question, usually referred to as the Nielsen Realization Problem, is whether every such finite subgroup arises as a group of isometries of some hyperbolic surface. In this paper we answer the question in the affirmative.

Groups Generated by reflections and aspherical manifolds not covered by Euclidean space

Abstract: A Coxeter system (r, V) is a group r (a "Coxeter group") together with a set of generators V such that each element of V has order two and such that all relations in r are consequences of relations of the form (VW)m(v, w) = 1, where v, w E V and m(v, w) denotes the order of vw. The m(v, w)'s (which are positive integers or oo) obviously determine the Coxeter system up to isomorphism. The list of finite Coxeter groups is short and well-known; however, general Coxeter groups are much more flexible. In fact, the set V and the m(v, w)'s can be specified arbitrarily (at least if V is finite) subject only to the conditions: m(v, v) = 1 and m(v, w) = m(w, v) ? 2 if v -# w. Suppose that (r, V) is a Coxeter system, that X is a Hausdorff space and that (Xv)v~v is a locally finite family of closed subspaces indexed by V. (The Xv are called the "panels" of X.(2)) There is a classical method for constructing a transformation group from these data. For each x E X, let V(x) denote the set of v in V such that x E Xv. For each subset S of V, let rs be the subgroup generated by S and let Xs be the "face" of X defined by

The structure of the Torelli group I : A finite set of generators for J

Abstract: This is the first of three papers concerning the so-called Torelli group. Let M = Mg be a compact orientable surface of genus g having n boundary components and let 9 = Xg . be its mapping class group, that is, the group of orientation preserving diffeomorphisms of M which are 1 on the boundary AM modulo isotopies which fix 3M pointwise. This group is also known to the complex analysts as the Teichmuller group or modular group. If n = 0 or 1, let further 4 = Jg . be the subgroup of D1 which acts trivially on H1(M, Z). The topologists have no traditional name for A, but the analysts tell me it was known classically and is called the Torelli group. Several interesting problems and conjectures exist concerning f. The principal one can be found in Kirby's problem list [K] and asks if gg is finitely generated. In this first paper we shall answer the question affirmatively for both gg 0 and fgg , when g > 3 and shall give a fairly simple set of generators. Two other conjectures were made by the author. The first involves the subgroup 'J of f which is generated by twists on nulhomologous simple closed curves. [JI] produces a surjective homomorphism T: fgg 1 A3H1(M, Z) which kills C, and it is conjectured there that in fact 5Y = Ker T. The proof of this is the content of the second paper. In the third paper we use the results of the second to compute the abelianization f/f' explicitly, thereby verifying another conjecture in [Ji]. The first reasonably simple (but infinite) set of generators for fg0 was produced by Powell in [P]. His generators consist of two types: a) twists on bounding simple closed curves, b) opposite twists on a (bounding) pair of disjoint homologous simple closed curves, each of which are nonbounding. Using his result, the author showed in [J2] that the maps of the second type, which we call BP maps (for bounding pair), are actually sufficient to generate both Kg 0 and fg 1 for g > 3 and in fact that we need only those whose two curves bound a genus one subsurface of M (note that for g = 2 all BP maps are trivial and hence the result fails in this case). In the finite set of generators produced in this paper only

A proof of the Smale Conjecture, Diff(S3) 0(4)

Abstract: The Smale Conjecture [9] is the assertion that the inclusion of the orthogonal group 0(4) into Diff(S3), the diffeomorphism group of the 3-sphere with the Cw topology, is a homotopy equivalence. There are many equivalent forms of this conjecture, some of which are listed in the appendix to this paper. We shall prove

An analogue of the prime number theorem for closed orbits of Axiom A flows

Abstract: For an Axiom A flow restricted to a basic set we extend the zeta function to an open set containing W(s) > h where h is the topological entropy. This enables us to give an asymptotic formula for the number of closed orbits by adapting the Wiener-Ikehara proof of the prime number theorem.

On the symplectic geometry of deformations of a hyperbolic surface

Abstract: Let R be a Riemann surface. In this manuscript we consider a geometry on the moduli space X(R) for R, which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of R. Classically the space of flat metrics for a torus is the locally symmetric space 0(2) \ SL(2; R)/SL(2; Z). We shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative Euler characteristic. The Teichmiiller space T(R), a covering of the moduli space, is a complex Kihler manifold. A KUhler metric for T(R), defined in terms of the Petersson product for automorphic forms, was introduced by Weil, [1]. The Weil-Petersson metric is invariant under the covering transformations and so projects to the moduli space X(R). The metric provides a link between the function theory of R and the geometry of X(R). In the Fenchel-Nielsen manuscript [8] a deformation, based on an amalgamation construction for Fuchsian groups, is introduced. The deformation is defined geometrically 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. The hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface. Choose a free homotopy class [a] on the surface R; then for each marked surface R realize [a] by the closed geodesic aR. The Fenchel-Nielsen deformations for the athen define a 1-parameter group of diffeomorphisms of T(R), whose infinitesimal generator by definition is the Fenchel-Nielsen vector field ta. In [21] the Fenchel-Nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t * was begun. The Fenchel-Nielsen vector fields were found to be related to the geodesic length functions 1*, introduced by Fricke-Klein to provide coordinates for T(R).

Necessary conditions for subellipticity of the $\overline\partial$-Neumann problem

Abstract: Let a be a a-closed form of type (p, q) with L2-coefficients on a smoothly bounded domain Q in C'. The problem of determining both the existence and regularity properties of the solution u of au = a, where u is orthogonal to the null space of a on (p, q l)-forms, is known as the a-Neumann problem. One of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates. Let U be a neighborhood of a point z0 in the boundary of Q. A subelliptic estimate is said to hold in U if the estimate

Unramified class field theory of arithmetical surfaces

Abstract: Let k be an algebraic number field, (9k its ring of integers and V a non-empty open subscheme of Spec(Ck). Let X be a projective smooth geometrically irreducible scheme over k, and X a regular proper flat scheme over V such that X x Vk X. The purpose of this paper is to give an explicit description of the abelian fundamental groups ,ab (X) and 7T b(X), using some "idele class groups" attached to X and X by a K-theoretical method. Its outline is as follows: In general, for a noetherian scheme Z and i = 0 or 1, we define the group SKi(Z) to be the cokernel of

Harmonic analysis on semigroups

Abstract: wherever t > 0 and 1 < p < so. E. M. Stein [St 3] developed a "LittlewoodPaley" theory for such semigroups, with some additional hypotheses. R. R. Coifman, R. Rochberg, and G. Weiss [CRW] pointed out that some of Stein's estimates could be obtained more simply by the use of "transference" techniques which were developed by Coifman and Weiss [CW]. Our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of Stein's book. We shall also use these results to prove estimates for the "wave equation"

Self-spreading and strength of singularities for solutions to semilinear wave equations

Abstract: Algebres microlocales. Propagation de la regularite. Solutions lineaires conduisant a l'etalement. Auto-etalement des singularites

Solution of the similarity problem for cyclic representations of C*-algebras

Abstract: In a paper from 1955 [13] Kadison considered the following problem: Is any bounded, non-selfadjoint representation 77 of a C*-algebra A on a Hilbert space H similar to a *-representation?; i.e. does there exist an invertible operator T E B(H), such that ThT( )T-' is a *-representation of A? This problem is a natural counterpart to a problem considered by Dixmier in 1950 [9]: Is any bounded representation of a group G on a Hilbert space similar to a unitary representation? The group question was solved in the negative by Kunze and Stein in 1960 [14], which in turn was based on previous work of Ehrenpreis and Mautner [21]. The similarity problem for representations of C*-algebras is still open, although a number of partial results have been obtained. Using the deep results of Connes on infective von Neumann algebras [7] and the characterization of nuclear C*-algebras given in [3], one gets the result fairly easily that every bounded representation of a nuclear C*-algebra is similar to a *-representation (cf. [2, Th. 3.5] and [5, Th. 4.1]). The similarity problem for bounded cyclic representations of arbitrary C*-algebras was first considered by Barnes in [1]. He proved that when 77 is a bounded representation of a C*-algebra A on a Hilbert space H. such that 7V(A) and v((A)* have a common cyclic vector I, then there exists a closed, infective operator T on H. such that T and T' are densely defined, and such that x -> ThT(x)T-' is a *-representation of A. Bunce sharpened this in [2] by proving that T can be chosen bounded, but not necessarily with bounded inverse. He also removed the condition that ( is cyclic for 7v(A)*. Finally, Christensen proved in [5] that every irreducible bounded representation of a C*-algebra is similar to a *-representation. These two results of Bunce and Christensen rely on an inequality due to Pisier [16, Cor. 2.3] stating that, when S is a bounded linear map from a C*-algebra A into a C*-algebra B. then for all n E N and a. an E A:

On the method of Thue-Siegel

Abstract: The methods of Thue and Siegel, based on explicit Pade approximations to algebraic functions, are used to examine diophantine approximations to roots of

Derivatives of analytic families of Banach spaces

Abstract: The theory of complex interpolation of Banach spaces and operators, which was developed by Calderon, Lions and S. G. Krein and extended by us and others, centers on the use of the maximum principle for analytic functions. In this paper we investigate the significance in interpolation theory of estimates for derivatives of analytic functions. We obtain norm estimates for certain linear and non-linear commutators and obtain new classical interpolation theorems. The paper has four sections. In the first section we show how Thorin's proof of the Riesz-Thorin theorem can be extended to give an estimate for a non-linear commutator and how analogous computations can be done in families of interpolation spaces. The second section introduces an extension of the complex interpolation theory which is designed to incorporate in a systematic way the type of estimates obtained in the first section. In the third section, the details of the abstract theory of the second section are filled in for several standard examples including LP spaces and weighted LP spaces. A brief final section contains some general comments and observations.

Group presentation, $p$-adic analytic groups and lattices in $\mathrm{SL}_2(\mathbf{C})$

Abstract: groups by using their pro-p completions. This method is based on the observation that if (X; R) is a presentation of the finitely generated (f.g.) group F, then (X; R), considered as a presentation for a pro-p group is actually a presentation for T,;, the pro-p completion of F. First we prove a generalization of a theorem of Golod and Shafarevich (cf. [H]). THEOREM. Let G be a p-adic analytic pro-p group different from Zp (= the p-adic integers). If (X; R) is a minimal presentation of G then 1X;2 IRI? 2

On virtual representations of symmetric spaces and their analytic continuation

Abstract: In this paper we study a problem in group representation theory motivated by relativistic quantum field theory. One of the most powerful approaches to constructing models of relativistic quantum fields relies on Euclidean field theory, a description of quantum field theory in which time is purely imaginary. The basic objects of Euclidean field theory (the Euclidean Green's- or Schwinger functions) can often be expressed in terms of functional integrals. The construction of quantum field models is thereby "reduced to quadratures". A well-known, simple example of this method is the reformulation of quantum mechanisms by means of Wiener integrals. (The fundamental solution of the Schrodinger equation, analytically continued to imaginary time, is given by the well-known Feynman-Kac formula; see e.g. [15], [19].) In this approach one faces the problem of analytic continuation back to real time. While this problem is elementary in non-relativistic quantum mechanics, it is rather intricate in relativistic quantum field theory. For some class of relativistic quantum field theories, a fairly general solution to this problem was given in papers by Osterwalder and