Showing papers in "Annals of Mathematics in 1981"

Spectral analysis of N-body Schrodinger operators *

Abstract: For a large class of two body potentials, we solve two of the main problems in the spectral analysis of multiparticle quantum Hamiltonians: explicitly, we prove that the point spectrum lies in a closed countable set (and describe that set in terms of the eigenvalues of Hamiltonians of subsystems) and that there is no singular continuous spectrum. We accomplish this by extending Mourre's work on three body problems to N-body problems.

On the Order of Uniprimitive Permutation Groups

Abstract: One of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group G of degree n, where G X An. We prove I G I < exp (4V'/ n log2 n) for the case when G is not doubly transitive. This result is best possible apart from an O(log n) factor in the exponent. The best result previously known was I G I < el (H. Wielandt). A similar estimate for the doubly transitive case follows in a subsequent paper. For rank 3 groups with subdegree p < n/2 we obtain I GI < exp (4(n/p) log2 n). In the proof we develop some new combinatorial properties of coherent configurations. The results also have relevance to theoretical estimates on the computational complexity of graph isomorphism testing.

The trace formula in invariant form

Abstract: Introduction The trace formula for GL2 has yielded a number of deep results on automorphic forms. The same results ought to hold for general groups, but so far, little progress has been made. One of the reasons has been the lack of a suitable trace formula. In [l(d)] and [l(e)] we presented a formula or, as we wrote it in [l(e), $51, G is a reductive group defined over Q, and f is any function in C;(G(A)l). The left hand side of (I*) is the trace of the convolution operator off on the space of cusp forms on G(Q) \\ G(A)'. It is a distribution which is of great importance in the study of automorphic representations. One would hope to study it through the distributions o E O} and { 1 ; : x E X \\ X(G)}. Unfortunately, these distributions depend on a number of unpleasant things. There is the parameter T , as well as a choice of maximal compact subgroup of G(A)' and a choice of minimal parabolic subgroup. What is worse, they are not invariant; their values change when f is replaced by a conjugate of itself. In any generalization of the applications of the trace formula for GL2, we would not be handed the function f. We could only expect to be given a function such as +(f 1: 77-t r d f 1 7 whose values are invariant in f. Here TT ranges over the irreducible tempered representations of G(A)'. The decomposition of trRcusp(f) into the right hand side of (l*) would then be of uncertain value, for the individual terms actually depend on f and not just +(f). a 1981 by Princeton University (Mathematics Department) For copying information, see inside back cover. The purpose of this paper is to modify the terms in (1) so that they are invariant. Under certain assumptions on the local groups G(Q_), we will obtain a formula in which the individual terms are invariant distributions. The definitions will be such that I = 1: if x belongs to X(G). We will therefore also have the analogue of (I*). The main assumptions on the local groups G (Q) are set forth in Section 5. One expects them to hold for all groups, but they are a little beyond the present state of harmonic analysis. They are, essentially, that any invariant distribution, I, on G(A)' can be identified with a …

A Property of the Functions and Distributions Annihilated by a Locally Integrable System of Complex Vector Fields

Abstract: page Introduction .387 1. Basic notation and ingredients.389 2. Approximation and constancy on the fibers of the classical solutions of the homogeneous equations . 393 3. First consequences and interpretation of the fundamental theorem. .398 4. Distribution solutions of the homogeneous equations. 402 5. Regularity of the solutions of the homogeneous equations .......409 6. Example of a system that is analytic hypo-elliptic but not Co hypo-elliptic.413 References 420

Hardy’s inequality and the $L^1$-norm of exponential sums

Abstract: In this paper we generalize Hardy's inequality [3] for measures of analytic type and obtain a proof of the Littlewood conjecture [4] for the L' norm of exponential sums as a simple consequence. Let T be the circle group, Z the additive group of integers and M(T) the customary convolution algebra of Borel measures on T; for pe e M(T) and n G Z put M(%)= e-ino dp(O) Denote by H'(T) the classical space of all measures e e M(T) such that j(n) = 0 for all n < 0 and let C denote the complex numbers. We now state Hardy's inequality for measures of analytic type: THEOREM 1. If M C H'(T) then

Varietes polaires locales et classes de Chern des varietes singulieres

Abstract: Dans ce travail, nous commencons l'etude systematique des varietes polaires locales attachees a un germe d'espace analytique complexe (que l'on peut decrire informellement comme lieux de points critiques non singuliers de projections generiques de cet espace sur des espaces affines). Nous etablissons un lien precis entre ces varietes polaires locales et la theorie des classes de Chern sur un espace singulier (due a R. MacPherson [24] et M. H. Schwartz [30]) en donnant pour l'obstruction d'Euler locale de R. MacPherson une expression algebrique comme somme alternee de multiplicites de varietes polaires. Nous tirons de ce resultat des methodes pour le calcul effectif des classes de Chern singulieres, une definition des classes de Chem locales et aussi des formules effectives pour la caracteristique d'Euler-Poincare x((V) d'une variete projective V, generalisant les formules de Plucker. Divers avatars du concept de variete polaire ont ete utilises depuis longtemps (Severi, Todd, Pontryaguine,... ) pour definir des invariants ("classes caracteristiques") des varietes algebriques projectives ou des varietes differentielles compactes. La relation entre varietes polaires et classes caracteristiques des varietes a ete estompee par l'extension de la theorie des classes caracteristiques aux fibres quelconques et le succes du point de vue axiomatique et cohomologique. Par ailleurs les varietes polaires elles-memes n'ont pas ete utilisees de fa~on vraiment geometrique puisque les geometres ne consideraient, meme apres l'extension au cas singulier en geometrie projective [28], que la classe d'equivalence rationnelle ou la classe de cohomologie des varietes polaires. Depuis plusieurs annees les auteurs du present travail ont ete amenes, dans l'etude locale des singularites des espaces analytiques, 'a utiliser des caracteres geometriques locaux de varietes polaires definies localement. Ces varietes interviennent aussi naturellement dans l'etude des singularites du point de vue

Degeneration of surfaces with trivial canonical bundle

Abstract: The object of this note is to prove the theorem stated below. The main idea of the proof, and much of the detail, are already to be found in V. I. Kulikov's important paper [K]. Our contribution to the proof, and the differences with Kulikov's paper are discussed later in this introduction. THEOREM. Let wl: X -> D be a semistable degeneration of surfaces such that:

The Dirichlet problem in non-smooth domains

Abstract: Dirichlet problem for an elliptic operator 2S- = laj jaj, with smooth coefficients aij, in domains D in Rm, m > 3, with non-smooth boundaries. The domains we treat are given locally, in some Cm coordinate system, by the graph of a continuous function qs, with Vq5 e LI, for some p, 1 ! p < oo. Such domains are called LI' domains. (See Section 1 for the precise definitions.) Note that L- domains are usually called Lipschitz domains. If D is an LI' domain, and X e D, we study the "elliptic" measure a7, associated with S and D at X. Thus, for f a continuous function on AD, the formula u(X) = Hf(X) f doH gives the Perron-Wiener-Brelot generalized soluID

Compact ergodic groups of automorphisms

Abstract: It is shown that if G is a compact ergodic group of *-automorphisms on a unital C*-algebra A then the unique G-invariant state is a trace. Hence if A is a von Neumann algebra then it is finite and infective.

On the radial behavior of minimal surfaces and the uniqueness of their tangent cones

Abstract: Suppose V is an m + 1-dimensional minimal surface in R" containing 0 as an isolated singular point. What can one say about the structure of V near O? This question initiated the present investigation and, in certain special cases, has a satisfactory answer. ( 1 ) If one of the tangent cones to V at 0 is of the form 0 * M where M is an m-dimensional minimal submanifold of S'-' and if also for each Jacobi normal vectorfield Z of M in Sn-1 there is a one parameter family of minimal surfaces in S'-1 having velocity Z at M then 0* M is the unique tangent cone to V at 0 with V converging to 0 * M for small radii r with rate rX', It > 0. ( 2 ) In case M is the cartesian product of two standard spheres of appropriate radii then each Jacobi vectorfield on M arises from isometric motions of Sn-1 and the Jacobi vectorfield hypothesis in (1) is satisfied. This is shown in Chapter 6. The uniqueness of tangent cones to one dimensional stationary varifolds with positive densities was shown in [AA], and the uniqueness of tangent cones to certain two dimensional area minimizing singular surfaces in R3 was shown by J. Taylor in [TJ1I, [TJ2], [TJ3]. Also B. White has shown such uniqueness for area minimizing hypersurfaces mod 4 in space dimen-

The topology of real algebraic sets with isolated singularities

Abstract: In this paper we give a topological classification of real algebraic sets with isolated singularities, showing that they are exactly smooth closed manifolds with smooth subpolyhedra crushed to points. The question of which topological spaces are homeomorphic to real algebraic sets (solutions of polynomial equations in Euclidean space) has been long studied. In 1936 Seifert showed that any smooth compact stably parallelizable manifold is diffeomorphic to a component of an algebraic set [121 and in 1952 Nash extended this result to all smooth compact manifolds [11]. In 1973 Tognoli showed that any smooth compact manifold is diffeomorphic to a nonsingular algebraic set [13], so at least compact nonsingular algebraic sets are classified. Little has been done with singular algebraic sets however, since the transversality arguments used by Seifert, Nash and Tognoli no longer apply except in some special cases. One could use stability of singularities such as Kuiper [71 and Akbulut [1] used to show certain nonsmoothable PL manifolds are algebraic sets or one could use the projective version of Seifert-Nash-Tognoli as King did [6], but one could still not hope these techniques would allow even a characterization of isolated singularities. To get around this problem we take a cue from Hironaka's resolution of singularities [4]. The idea is to take a 'topological resolution' of a space if it exists. We can apply transversality techniques (SeifertNash-Tognoli) to the resolved space and then blow down algebraically and end up with the original space as an algebraic set. It seems likely that this technique allows one to classify all algebraic sets but in any case, we show that it classifies all algebraic sets with isolated singularities. In future papers we will use this technique to show, for instance, that all compact PL manifolds are homeomorphic to real algebraic sets [17] and that 2-dimensional real algebraic sets are topologically characterized as polyhedra satisfying Sullivan's even local Euler characteristic condition [16].

Kohn-Rossi Cohomology and its Application to the Complex Plateau Problem, I

Abstract: Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension 2n-1 in $\\mathbb{C}^{N}$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\\ge 3$ and $N=n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn-Rossi cohomology groups on $X$ in 1981. For n=2 and $N\\ge n+1$, the problem has been open for over 30 years. In this paper we introduce a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. In case $n=2$ and N=3, the vanishing of this invariant is enough to give the interior regularity.

The Gauss Map of a Complete Non-Flat Minimal Surface Cannot Omit 7 Points of the Sphere

Abstract: A well-known theorem of Osserman ([21, [3], [4]) states that the Gauss map of a complete minimal surface M2 c R3 cannot omit a set of positive logarithmic capacity unless the surface is a plane. In this paper we improve Osserman's theorem by showirg that the Gauss map of M2 omits at most 6 points (provided M2 is not flat). It should be pointed out, however, that no example is known where the omitted set has 5 points. Therefore the problem of determining the exact size of the omitted set remains unsolved (we refer to [2] and [31 for a historical account as well as for general facts pertaining to minimal surfaces). We take the opportunity to thank L. Petrola for sharing his insights with us and Professor J. Shah for some very enlightening conversations on related matters.

An ordinal $L^p$-index for Banach spaces, with application to complemented subspaces of $L^p$

Abstract: One of the central problems in the Banach space theory of the LP-spaces is to classify their complemented subspaces up to isomorphism (i.e., linear homeomorphism). Let us fix 1 < p < xc, p =# 2. There are five "simple" examples, LP, UP, 12, 12 @ Up, and (12 @ 12 @ ... )P. Although these were the only infinitedimensional ones known for some time, further impetus to their study was given by the discoveries of Lindenstrauss and Pelczyniski [15] and Lindenstrauss and Rosenthal [16]. These discoveries showed that a separable infinite-dimensional Banach space is isomorphic to a complemented subspace of LP if and only if it is isomorphic to 12 or is an "EP-space", that is, equal to the closure of an increasing union of finite-dimensional spaces uniformly close to 1'P's. By making crucial use of statistical independence, the second author produced several more examples in [19], and the third author built infinitely many non-isomorphic examples in [23]. These discoveries left unanswered: Does there exist a Xp and infinitely many non-isomorphic Xp-complemented subspaces of LP (equivalently, are there infinitely many separable Ep A-spaces for some X depending on p)? We answer these questions by obtaining uncountably many non-isomorphic complemented subspaces of LP.* Before our work, it was suspected that every EP-space nonisomorphic to LP embedded in (12e 12e ... )P (for 2 < p < oc) (see Problem 1 of [23]). Indeed, all the known examples had this property. However our results show that there is no universal fp-space besides LP. To obtain these results, we use rather deep properties of martingales together with a new ordinal index, called the local LP-index, which assigns "large" countable ordinals to any

On certain zeta functions attached to two Hilbert modular forms. I. The case of Hecke characters

Abstract: To make our exposition smooth, let us first introduce some notational conventions. For an algebraic number field K of finite degree, we denote by JK the set of all embeddings of K into C, and by IK the free Z-module generated by the elements of JK. We then put RIK = IK0 R and CIK = IK0 C. If p = EaPaU E IK with a E JK and PO E Z. we put xP = fl,(xa)Pfor 0 / x E K; this is meaningful for p e CIK if xa are all real and positive. Throughout the paper, we denote by D the complex upper half plane and by F a totally real algebraic number field of degree n. Now the zeta function to be studied in this paper, when suitably specialized, has the form

On certain zeta functions attached to two Hilbert modular forms II. The case of automorphic forms on a quaternion algebra

Abstract: As to the meaning of the symbols, the reader is referred to the introduction of Part I. We recall here only that F is a totally real algebraic number field of degree n, and w denotes the Fourier coefficients of an elliptic modular form 52(z) = E wQ(a)e2q'iaz. In Part I, we investigated D assuming that EX(a)N(Q) S is an L-function of a CM-field with an algebraic-valued Hecke character. In the present Part II, we treat the case where X( a) can be obtained as eigenvalues of Hecke operators T(a) on a quaternion algebra B over F. Thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane & with respect to congruence subgroups of B x . Here r is the number of archimedean primes of F unramified in B. Then we introduce in Section 2 the notion of arithmeticity (or Q-rationality) of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones. Our main theorems will be stated in Section 3 and proved in Section 6; Sections 4 and 5 are preliminaries to the proof. The central result, specialized to the case r = 1, asserts that if g is a Q-rational eigenform such that gj T( a) =X( a )g for all integral ideals a of F, then D(s0) for certain integers so are algebraic numbers times 7Tdg, g), where d is an integer determined by the weight of g, and (*, *) is the Petersson inner product.