Showing papers in "Annals of Mathematics in 1996"

Global uniqueness for a two-dimensional inverse boundary value problem

Abstract: We show that the coefficient -y(x) of the elliptic equation Vie (QyVu) = 0 in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a reconstruction pro

Lower bounds on Ricci curvature and the almost rigidity of warped products

Abstract: The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the "volume cone implies metric cone" theorem, the maximal diameter theorem, [Cg], and the splitting theorem, [CG]. Each asserts that if a certain geometric quantity (volume or diameter) is as large as possible relative to the pertinent lower bound on Ricci curvature, then the metric on the manifold in question is a warped product metric of a particular type. In this paper we provide quantitative generalizations of the above mentioned results. Among the applications are the splitting theorem for GromovHausdorff limit spaces X, where Mn -* X, Ricmn ? -i see [FY]. Other applications include the assertion that for complete manifolds, M', with Ricmf > 0 and Euclidean volume growth, all tangent cones at infinity are metric cones; compare [BKN], [CT], [P1]. Via resealing arguments, there are also strong consequences for the local structure of manifolds whose Ricci curvature satisfies a fixed lower bound and for their Gromov-Hausdorff limits. Some of these are announced in [CCol]; for a more detailed discussion see [CCo2], [CCo3], [CCo4]. Our work further develops and significantly extends techniques which were introduced in [Col], [Co2] and significantly extended in [Co3], in order to prove certain "stability" conjectures of Anderson-Cheeger, Gromov and Perelman. The results of [Col]-[Co3] were announced in [Co4]. We briefly review some of those results. Let dGH denote the Gromov-Hausdorff distance between metric spaces; see [GLP]. Let S' denote the unit sphere and recall that S' is the unique complete

Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation

Abstract: *This paper was written while both authors were guests of the Forschungsinstitut fur Mathematik at the ETH Zurich, and we thank the institute for its hospitality, pleasant working atmosphere and helpful staff. In particular, we thank Jiirgen Moser for numerous stimulating discussions on the subject. We also benefitted from a remark by Sigurd Angenent concerning the analyticity of the solutions. The second author also thanks the Deutsche Forschungsgemeinschaft for their financial support through a Heisenberg grant.

The Cauchy integral, analytic capacity, and uniform rectifiability

Abstract: Several explanations concerning notation, terminology, and background are in order. First notation: by 7Hi we have denoted the one-dimensional Hausdorff measure (i.e. length), and A(z,r) stands for the closed disc with center z and radius r. A curve F is called AD-regular, that is, Ahlfors-David-regular, if it satisfies (1.2) (with E = F). Since the lower bound is automatic for curves, this means that 'Hl (r n A(z, r)) 0. General sets satisfying (1.2) are called AD-regular. It is simplest to define the L2-boundedness of the Cauchy singular integral operator via the truncated integrals: we say that CE is bounded in L2(E) (without really defining the operator CE itself) if there is M < ox such that

A characterization of ideal polyhedra in hyperbolic $3$-space

Abstract: The goals of this paper are to provide a characterization of dihedral angles of convex ideal (those with all vertices on the sphere at infinity) polyhedra in H3, and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of H3. These characterizations are given in, respectively, Theorems 0.1 and 10.5. The first theorem is proved in detail, while the proof of the second (which is similar) is only outlined. The results of this paper grow out of the general framework of the author's doctoral dissertation [13], as published in [20]. A lot of the language, and some of the auxilary results come from there as well, so familiarity with the latter reference is very helpful. The necessity of the conditions postulated in Theorem 0.1 has been shown in [14]; hence only the sufficiency direction will be shown in the current paper. In 1832, Jakob Steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere. This was considered intractable it took almost a hundred years to find a single example of an "uninscribable" combinatorial type (by Steinitz [24]). However, Theorem 0.1 gives such a characterization. Section 11 is dedicated to a (very brief) historical survey and some purely graph-theoretic and computational-geometric consequences of Theorem 0.1. In order to state this theorem, suppose that a convex ideal polyhedron P in H3 is given. Let P* denote the Poincar' dual of P, and assign to each edge e* of P* a weight w(e*) equal to the exterior dihedral angle at the corresponding edge e of P. Then the following result holds:

Embedding problems over large fields

Abstract: In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on varieties. As applications, we give a positive answer to a long standing conjecture which originates in an unpublished note of Roquette, and asserts that the absolute Galois group of a countable, hilbertian, PAC field is profinite free; see [F-J, Problem 24.41]. Secondly, we give new evidence for the conjecture of Shafarevich which asserts that the absolute Galois group of Qab is profinite free. Finally, one of the most interesting applications of the theory we develop here is the insight in the Galois structure of the totally EG-adic numbers.

Actions propres sur les espaces homogenes reductifs

Abstract: Let G be a linear semisimple real Lie group and H be a re- ductive subgroup of G. We give a necessary and sufficient condition for the existence of a nonabelian free discrete subgroup r of G acting properly on G/H. For instance, such a group r does exist for SL(2n, IR)/SL(2n - 1, IR) but does not for SL(2n + 1, R)/SL(2n, R) with n > 1.

Classical groups, probabilistic methods, and the (2,3)-generation problem

Abstract: We study the probability that randomly chosen elements of prescribed type in a finite simple classical group G generate G; in particular, we prove a conjecture of Kantor and Lubotzky in this area. The probabilistic approach is then used to determine the finite simple classical quotients of the modular group PSL2(Z), up to finitely many exceptions.

Irreducible components of the space of holomorphic foliations of degree two in cp(n), n 3

Abstract: In this paper we will prove that the space of holomorphic fo- liations of codimension 1 and degree 2 in CP(n), n > 3, has six irreducible components.

Topology of homology manifolds

Abstract: ANR homology n-manifolds are nite-dimensional absolute neighborhood retracts X with the property that for every x 2 X, Hi(X;X fxg) is 0 for i 6= n and Z for i = n. Topological manifolds are natural examples of such spaces. To obtain nonmanifold examples, we can take a manifold whose boundary consists of a union of integral homology spheres and glue on the cone on each one of the boundary components. The resulting space is not a manifold if the fundamental group of any boundary component is a nontrivial perfect group. It is a consequence of the double suspension theorem of Cannon and Edwards that, as in the examples above, the singularities of polyhedral ANR homology manifolds are isolated. There are, however, many examples of ANR homology manifolds which have no manifold points whatever. See [12] for a good exposition of the relevant theory. The purpose of this paper is to begin a surgical classi cation of ANR homology manifolds, sometimes referred to in the sequel, simply as homology manifolds. One way to approach this circle of ideas is via the problem of characterizing topological manifolds among ANR homology manifolds. In Cannon's work on the double suspension problem [6], it became clear that in dimensions greater than 4, the right transversality hypothesis is the following (weak) form of general position.

Cases of equality in the riesz rearrangement inequality

Abstract: We determine the cases of equality in the Riesz rearrangement inequality ZZ f (y)g(x ? y)h(x) dydx ZZ f (y)g (x ? y)h (x) dydx where f , g , and h are the spherically decreasing rearrangements of the functions f , g, and h on R n. We apply our results to the weak Young inequality. The Riesz rearrangement inequality states that the functional I(f; g; h) := Z f gh dx = ZZ f(y)g(x?y)h(x) dydx (1:1) never decreases under spherical rearrangement, that is, 1 for any triple (f; g; h) of nonnegative measurable functions on R n for which the right hand side is deened. The spherically decreasing rearrangement, f , of a nonnegative measurable function f is the spherically decreasing function equimeasurable to f. We will deene it by f (x) = sup n s > 0 j (N s (f)) ! n jxj n o ; where N s (f) := n x 2 R n j f(x) > s o is the level set of f at height s, and ! n denotes the measure of the unit ball in R n. That is, the level sets of f are the centered balls of equal measure as the corresponding level sets of f. This deenition makes sense if all level sets corresponding to positive values of f have nite measure, for example, if f is in L p for some p < 1. In this paper, we determine the cases of equality in (1.2). A triple of functions that satisses (1.2) with equality will be called an optimizing triple, or optimizer, of the inequality. There are many optimizers of (1.2). One reason is that I is invariant under a large group of aane transformations: For any linear map, L, of determinant 1, and vectors a, b, and c = a + b in R n , we have where g ? denotes the function deened by g ? (x) := g(?x). Clearly, any triple of functions that is equivalent to a triple of spherically decreasing functions under these symmetries is an optimizer. There is a second reason to expect many optimizers. Consider the case when f and g have compact support. Then also the convolution f g has compact support. If h is the characteristic function of a set that contains the support of f g, then f; g; h produce equality in (1.2) regardless …

The blowup formula for donaldson invariants

Abstract: k k! which is calculated in §4 below. This formula has been the target of much recent work. The abstract fact that there exists such a formula which is independent of X was first proved by C. Taubes using techniques of (T). J. Bryan (B) and P. Ozsvath (O) have independently calculated the coefficients throughB10(x). Quite recently, J. Morgan and Ozsvath have announced a scheme which can recursively compute all of the Bk(x). The special case of the blowup formula for manifolds of "simple type" (see §5 below) was first given by P. Kronheimer and T. Mrowka. However, none of the techniques in these cases approach the simplicity of that offered here. Before presenting the formula, we shall first establish notation for the Donaldson in- variants of a simply connected 4-manifold X with b + > 1 and odd. (The hypothesis of simple connectivity is not necessary, but makes the exposition easier.) An orientation of X, together with an orientation of H 2 + (X; R) is called a homology orientation of X. Such a homology orientation determines the (SU(2)) Donaldson invariant, a linear function D = DX : A(X) = Sym∗(H0(X) ⊕ H2(X)) → R

$L^2$ solvability and representation by caloric layer potentials in time-varying domains

Abstract: We consider boundary value problems for the heat equation in time varying graph domains of the form Ω = {(x0, x, t) ∈ IR × IRn−1 × IR : x0 > A(x, t) }, obtaining solvability of the Dirichlet and Neumann problems when the data lies in L(∂Ω). We also prove optimal regularity estimates for solutions to the Dirichlet problem when the data lies in a parabolic Sobolev space of functions having a tangential (spatial) gradient, and 1/2 of a time derivative in L(∂Ω). Furthermore, we obtain representations of our solutions as caloric layer potentials. We prove these results for functions A(x, t) satisfying a minimal regularity condition which is essentially sharp from the point of view of the related singular integral theory. We construct counter examples which show that our results are in the nature of “ best possible. ” 1991 Mathematics Subject Classification. Primary 42B20, 35K05. keywords and phrases. heat equation, Dirichlet problem, Neumann problem, layer potentials, timevarying domains, singular integrals, Rellich inequalities. 1 Supported by an NSF Grant 0. Background and Notation. A longstanding problem concerning solvability of the Dirichlet problem for Laplace’s equation in a Lipschitz domain was resolved by B. Dahlberg [D1], who showed that in such domains harmonic measure, dω, and surface measure, dσ, are mutually absolutely continuous, and furthermore, that the Dirichlet problem is solvable with data in L(dσ) (and consequently with data in L, 2− < p <∞). R. Hunt proposed the problem of finding an analogue of Dahlberg’s result for the heat equation in domains whose boundaries are given locally as graphs of functions A(x, t) which are Lipschitz in the space variable. It was conjectured at one time that A should be Lip 1 2 in the time variable, but subsequent counterexamples of Kaufmann and Wu [KW] showed that this condition does not suffice. Motivated in part by work of Strichartz [Stz] on BMO Sobolev spaces, and in part by work of M. Murray [Mu], Lewis and Murray [LM], made significant progress toward a solution of Hunt’s question, by establishing mutual absolute continuity of caloric measure and a certain parabolic analogue of surface measure in the case that A has 1 2 of a time derivative in BMO(IR) on rectangles, a condition only slightly stronger than Lip 1 2 . Furthermore these authors obtained solvability of the Dirichlet problem with data in L, for p sufficiently large, but unspecified. The regularity condition which Lewis and Murray imposed upon A(x, t) (or, to be more precise, an equivalent formulation of it) was shown by the first named author to be necessary and sufficient for L boundedness of the first parabolic Calderón commutator, thus further clarifying the connection between the results of [LM] and those of [D1]. Still, by analogy to [D1], it remained an open problem to treat the case of boundary value problems with L data in the parabolic setting. It is this issue of L solvability that we address here. To be more specific in this paper we study the Dirichlet and Neumann problems for the heat equation in non cylindrical (i.e. time-varying) graph domains. We treat each of these problems in the case that the data belongs to L with respect to a certain projective Lebesgue measure. We 1 also consider regularity estimates for solutions of the Dirichlet problem when the data belongs to a parabolic Sobolev space having a full spatial derivative and one half of a time derivative in L. Existence of our solutions will be obtained by using the method of layer potentials. In addition we shall give an alternate, simpler proof of recent results of the first author [H2] concerning “ smoothing operators of Calderòn type, ” including the caloric single layer potential. We shall study these problems in graph domains of the form Ω = {(x0, x, t) ∈ IR× IRn−1 × IR : x0 > A(x, t) } (0.1) where n ≥ 2 and A(x, t) is Lipschitz in the space variable, uniformly in time, i.e., |A(x, t)− A(y, t)| ≤ β0 |x− y|, x, y ∈ IRn−1, t ∈ IR, (0.2) and where A(x, t) satisfies a certain half order smoothness condition in the time variable. To describe this condition we follow Fabes and Riviere [FR1] and define a half-order time derivative by IDnA(x, t) = ( τ ‖(ξ, τ)‖ Â(ξ, τ) )̌ (x, t) (0.3) whereˆandˇdenote respectively the Fourier and inverse Fourier transforms on IR, and ξ, τ denote, respectively, the space and time variables on the Fourier transform side. Also ‖z‖ denotes the parabolic “ norm ” of z. We recall that this “ norm ” satisfies the non-isotropic dilation invariance property ‖(δx, δt)‖ ≡ δ‖(x, t)‖. Indeed, ‖(x, t)‖ is defined as the unique positive solution ρ of the equation n−1 ∑ i=1 xi ρ2 + t ρ4 = 1. (0.4) The half order smoothness condition in the time variable which we impose upon A is that IDnA ∈ (parabolic) BMO. We recall that parabolic BMO is the space of all locally integrable 2 functions modulo constants satisfying ‖b‖∗ ≡ sup B 1 |B| ∫ B |b(z)−mBb| dz < ∞. (0.5) Here, z = (x, t) and B denotes the parabolic ball B ≡ Br(z0) ≡ {z ∈ IR : ‖z − z0‖ < r} (0.6) where |B| denotes the Lesbegue n measure of B and mBb ≡ 1 |B| ∫ B b(z)dz. We note that |Br(z0)| ≡ cr where c is a constant and d = n + 1 is the homogeneous dimension of IR endowed with the metric induced by ‖ · ‖. We observe that IR so endowed is a space of homogeneous type in the sense of Coifman and Weiss [CW]. Indeed, there is a polar decomposition z ≡ (x, t) ≡ (ρθ1, . . . , ρθn−1, ρθn), dz ≡ dxdt ≡ ρd−1(1 + θ n)dρ dθ (0.7) where θ = (θ1, . . . , θn), |θ| = 1, and dθ denotes surface area on the unit sphere. Throughout this paper Lp(IRn−1), 1 < p < ∞, denotes, as usual, the space of p th power integrable functions f on IRn−1 with norm, ‖f‖p . To explain the significance of the conditions which we have imposed upon A, we recall a result of the first author [H1], which states that ∥∥∥∥∥ ∆− ∂ ∂t , A  ∥∥∥∥∥ op ≈ ‖∇xA‖∞ + ‖IDnA‖∗, where ≈ means the two quantities are bounded by constant multiples of each other. Moreover, ‖ · ‖ denotes the operator norm on L2(IRn−1), and ∇x ≡ ( ∂ ∂x1 , . . . , ∂ ∂xn−1 ). (0.8)

Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems

Abstract: in some domain D C Rn+l that vanishes locally on some distinguished part of OD. The term refers to the fact that in some sense, along that part of OD, u resembles the fundamental solution of the heat equation in D. This is particularly the case when the domain under consideration, D, is the intersection of some n+1-dimensional cube Q with Q, one side of a Lipschitz graph; i.e., Q = {fx > f(x', t)} (that is D = Q n Q) and the distinguished part of OD is precisely (&Q) n Q. This is the main first area of study of this paper, relying on work of Fabes, Garofalo, Salsa [FGS] on backward Harnack type inequalities for such domains. Lipschitz regularity in time, versus Lipschitz regularity in space, is not, of course, the natural homogeneity balance for the study of parabolic equations, but it is so for the study of phase transition relations of the form

Del Pezzo surfaces over Dedekind schemes

Abstract: Let S be a Dedekind scheme with fraction field K. We study the following problem: given a Del Pezzo surface X, defined over K, construct a distinguished integral model of X, defined over all of S. We provide a satisfactory answer if S is a smooth complex curve, and a conjectural answer if X is a cubic Del Pezzo surface over (nearly) arbitrary S.

Eigenvalues of the laplacian, the first betti number and the congruence subgroup problem

Abstract: THEOREM 1.1. Let F be an arithmetic lattice in the real Lie group SO(n,1), n > 2. (If n = 7 then assume r does not come from the twisted forms 3'6D4. If n = 3 and F comes from the units of a quaternion algebra over a number field L with a unique complex embedding, then assume L has a subfield of index 2.) Then for some finite index congruence subgroup Fo of r, the commutator quotient is infinite. Equivalently, 1(Fo), the first Betti number of the manifold FO \ Hn, is nonzero when 1 = SO(n,1)/SO(n) is the n-dimensional hyperbolic space.

A note on the Gauss-Bonnet theorem for Finsler spaces

Abstract: Exactly fifty years ago, one of us gave a proof of the Gauss-Bonnet formula for Riemannian manifolds by the method of transgression ([Chl], [Ch2]), and introduced a 'total curvature' H whose properties have yet to be fully exploited. Other proofs have since been given, including one as the simplest case of the Atiyah-Singer index theorem. The Finsler side of the story is what concerns us in the following pages, and it begins with a work of Lichnerowicz's [L] in 1948. In that paper, using the Cartan connection, Lichnerowicz established a Gauss-Bonnet theorem for all Finsler surfaces [modulo the issue of Vol(x) discussed below] and also for all Cartan-Berwald spaces of even dimension greater than two. His proof was modelled after the intrinsic method just mentioned. There are several interesting issues raised by Lichnerowicz's paper. One concerns the volume Vol(x) of the unit Finsler sphere IxM in each tangent space TxM. Various attempts to understand why he assigned these volumes the constant Euclidean values (as in Riemannian geometry) have led to some developments which play a key role in our treatment here. There is also the issue which revolves around the choice of a connection. It appears that his restriction to Cartan-Berwald spaces was dictated by the structure of the curvature tensor of the Cartan connection. We will show that the use of a connection introduced by one of us [Ch3] in 1948 effects the extension of Lichnerowicz's result to a much larger class of Finsler manifolds. Finally, there is the question of where the Gauss-Bonnet integrand should live. In Riemannian geometry, it is a top degree form on the underlying manifold M. It was Lichnerowicz who proposed that for the Finsler case, little is lost by allowing this integrand to live on the projective sphere bundle SM, as

Noncompact simple automorphism groups of Lorentz manifolds and other geometric manifolds

Abstract: When such a G does act, Zimmer has proven that the full automorphism group of the Lorentz manifold must be locally isomorphic to SL(2, R) x K, where K is a compact Lie group. M. Gromov has shown [2] that the universal cover M must then be Lorentz isometric to a warped product SL(2, R)f x S, where SL(2,IR) is the universal cover of SL(2, II) with the Cartan-Killing metric, S is a Riemannian manifold, and f: S -(0,O. +oo) is a warping function. This paper is mostly concerned with the following situation: a noncompact simple R-algebraic group G acting on a manifold M preserving a Lorentz metric. We do not assume M compact or finite volume or even complete; in contrast, the theorems above require compactness and use it in an essential way in their proofs. But, as we show, strong analogues of these theorems hold even for M noncompact. Our first result is the following:

On the fundamental groups of manifolds of positive sectional curvature

Abstract: Let M be an n-manifold with sectional curvature 0 < 6 < K < 1. The main result asserts that the fundamental group of M has a finite normal cyclic subgroup with index less than w(n, 8) < oc, a constant depending on n and 8.

Solvability for a class of doubly characteristic differential operators on $2$-step nilpotent groups

Abstract: Since H. Lewy's [Le] discovery of a smooth linear partial differential operator without solution, the problem of solvability of linear differential operators has been investigated intensively by various authors. This has meanwhile led, most notably through the work of Hdrmander, Maslov, Egorov, Nirenberg~erves, and Beals-Fefferman, but with many important contributions also by others, to a very satisfactory theory in the case of principal type operators (see [H6]). Much less is known about operators whose principal symbol vanishes of second order on (part of) its characteristic variety; i.e., those which admit double characteristics, even in the case of real principal symbols. The main objective of this article is to present a "model class" of such operators for which we are able to discuss solvability in a complete way. Our class, being composed of rather special operators, will certainly not capture all phenomena which might arise in the study of general doubly characteristic operators with real symbol, as the results in [St] already indicate, but it is still rich enough to exhibit several new features, if one compares with operators of principal type. The operators that we shall consider will be left-invariant differential operators on connected, simply connected 2-step nilpotent Lie groups G of the form m (0.1) LE ajkVjVk + U. j,k=l

Meromorphic continuation of L-functions of p-adic representations

Abstract: x det(I -Td(x)B(Xqd(x)-l) ...B(Xq)B(x))' where x is the Teichmiiller lifting of Zt and d(x) denotes the degree of x over Fq. This infinite product is a well-defined power series with coefficients in Rp because there are only finitely many closed points of a given degree. If V/Fq is an affine variety contained in An, then one defines L(B/V, T) to be the product in equation (1.1) except that we now restrict x to run only over the closed points of V/Fq. Using a standard reduction argument and Dwork's

Dimension and rigidity of quasi-Fuchsian representations

Abstract: Let FO C SO(n, 1) (n > 2) be a cocompact lattice and p: Fo - F be an injective representation into a convex-cocompact discrete isometric subgroup of a noncompact rank-1 symmetric space. The Hausdorff dimension 6(F) of the limit set of F = p(Fo) satisfies 6(F) > 6(Fo) = n - 1. We prove that equality holds if and only if p is a Fuchsian representation; i.e., F preserves a totally geodesic copy of HRI in HR. This generalizes the result of (2) and settles a question raised by Tukia ((43), p. 428). Actually we prove a more general result in the context of variable negative curvature. Strikingly there are no quasi-Fuchsian representations at least for the lower codimensional case in complex hyperbolic geometry. That is, for a cocompact lattice FO C SU(n, 1) (n > 2) and an injective representation p: Fo -* SU(m, 1) (n < m < 2n - 1) with F = p(Fo) convex-cocompact, we prove that one always has 6(F) = 6(Fo) and moreover, F must stabilize a totally geodesic copy of H?: in H?:. This can be viewed as a global generalization of Goldman and Millson's local rigidity theorem (see (20); another global generalization was obtained by K. Corlette (5)). Various other related rigidity results are also obtained.

Divergence of decreasing rearranged Fourier series

Abstract: There exists a square integrable function whose Fourier sum, when taken in decreasing order of magnitude of the coefficients, diverges unboundedly almost everywhere.