A geometrical interpretation of the shape invariant for geodesic triangles in complex projective spaces

We show that the shape invariant of a triangle in the complex projective space P ⁿ , see [B], can be obtained by integrating the Kählerian form of P ⁿ over a domain parametrized by geodesics and bounded by a geodesic loop formed with sides of the triangle.The second author was supported by a grant from INDAM-Rome.     

A note on the classification of holomorphic harmonic morphisms

In this note we give a complete classification of those holomorphic maps :U ⁿ defined on open and connected subsets of ^m which are harmonic morphisms.The first author was supported by the Icelandic Science Fund.     

Renormalization of cocycles and linear ODE with almost-periodic coefficients

Summary In the current paper we address the problem of classification of cocycles over an irrational rotation. We use the renormalization group approach.A cocycle means aC ^r -mappingu:T SU (2, ) (r2). We fix an irrational rotation :T T. Two cocyclesu, v:T SU(2, ) are considered equivalent (or cohomologous) if there is a continuous maph:TSU (2, ) such thatu·h =h·v. By definition, our problem is to classify cocycles up to this equivalence relation.By introducing a suitable renormalization map we are able to define a notion of afiber rotation number for a class of cocycles which are in the basin of the attractor of the renormalization map. The attractor itself is built of algebraic Anosov maps onT ². We present a number of results and conjecuters resulting from this approach.We show how this approach sheds some light upon the problem of classifying linear ODE with almost-periodic, skew-hermitian coefficient matrix.Oblatum 29-VIII-1990 & 4-III-1992A large part of this paper was written during the author's stay at the Institute for Advanced Study. This is version 1.26 of 2/27/92     

Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

We show that the bi-Lipschitz equivalence of analytic function germs (², 0)(, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families f _t : (², 0)(, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.     

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

Totally real imbeddings and the universal covering spaces of domains of holomorphy: Some examples

We begin with a review of the known examples of compact totally realn-dimensional submanifolds of _n . We then construct some new families of examples, including some which are simply connected. We conclude by using these examples to construct bounded domains of holomorphy in _n whose universal covering spaces are not biholomorphically equivalent to domains in _n .Herrn Prof. Dr. Karl Stein gewidmetResearch supported in part by Grant MCS 8301142 from the National Science FoundationResearch supported in part by Grant MCS 8219229 from the National Science Foundation     

Parabolic exhaustions for strictly convex domains

Using results of L. Lempert, we are able to construct parabolic exhaustions for strictly convex domains D^m with center at any given point of D. Using the theory of parabolic spaces and the geometric properties of these exhaustions, we can characterize the strictly convex domains biholomorphic to a circular domain and in particular to the ball in ^m.Supported by a grant from the C.N.R. (Italy)     

On the quotient ring by diagonal invariants

For a finite Coxeter group, W, and its reflection representation , we find the character and Hilbert series for a quotient ring of [^*] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman.     

Representation of distributions with compact support

Distributions having compact support are represented as the boundary value of Cauchy and Poisson integrals corresponding to tubular radial domains T^C in ⁿ where C is an open convex cone. The Cauchy integral of U is shown to be an analytic function in T^C which satifies a certain boundedness condition. All analytic functions in T^C having this boundedness condition have a distributional boundary value which can be used to determine an distribution. The results are extended to vector valued distributions.     

Linking Class of Lagrangian or Totally Real Embeddings

Given a totally real embedding j of the 2-torus into ², one defines a 1-class ₁ – its linking class – which is a tool to detect arcwise connected components of the space of totally real embeddings EmbTr( , ²). We generalize the construction of the linking class to any totally real embedding j of a connected, oriented, compact manifold without boundary M ⁿ into ⁿ. We obtain an (n – 1)-class _{n– 1} which is still an invariant for isotopy classes of totally real embeddings. We show that this class is nontrivial by computing it for some families of totally real embeddings. We then study the relationship between isotopy classes of ordinary embeddings and the linking class. With additional assumptions on M ⁿ (n 4 and M ⁿ parallelizable) we obtain the following: two totally real embeddings of M ⁿ into ⁿ which belong to the same isotopy class of totally real immersion, belong to the same isotopy class of ordinary embedding if and only if (1) their linking classes are the same (if n odd); (2) the images of their linking classes by the coefficient homomorphism : H ^{n– 1} (M ⁿ , ) H ^{n– 1} (M ⁿ , ₂) are the same (if n even).     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

New examples of Kobayashi hyperbolic surfaces in ℂℙ3

We show that a generic small deformation of the union of two generic cones in ³ of degree 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in ³ of any given degree d 8.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 90–94, 2005Original Russian Text Copyright © by B. Shiffman and M. ZaidenbergThe second authors research was supported in part by NSF grant No. DMS-0100474.Translated by B. Shiffman and M. Zaidenberg     

On the Fredholm theory of Wiener-Hopf equations and the coupling method

The Fredholm properties of the Wiener-Hopf operator onL _p(⁺,^m) are investigated using the coupling method for solving operator equations. The theory applies to equations whose kernel is an element ofL ₁(,^mxm). As usual the determinant of the symbol is assumed to have no zeros on the real line. The method of analysis is independent of the realization theory for symbols that are analytic in a strip containing the real axis although in some sense closely related to it. The connection between the two methods is briefly analysed in the paper.     

Ergodic Properties of Function Groups

Let PSL(2,) be a torsion free function group. We discuss some aspects of the asymptotic relation between the geometry of the hyperbolic manifold H³/ and the geometry of the boundary of its convex core. This leads to a characterization of Fuchsian function groups by dynamical invariants.     

On the dimensions of automorphism groups of eight-dimensional ternary fields,II

LetT be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed fieldF of is connected, then is either isomorphic to one of the compact Lie groupsG ₂ or SU₃, or the (covering) dimension of is at most 7.     

Surjectivity of quotient maps for algebraic (ℂ,+)-actions and polynomial maps with contractible fibers

In this paper we establish two results concerning algebraic (,+)-actions on ⁿ . First, let be an algebraic (,+)-action on ³. By a result of Miyanishi, its ring of invariants is isomorphic to [t ₁,t ₂]. Iff ₁,f ₂ generate this ring, the quotient map of is the mapF:³²,x(f ₁(x), f₂(x)). By using some topological arguments we prove thatF is always surjective. Secon, we are interested in dominant polynomial mapsF: ^{n n-1} whose connected components of their generic fibers are contractible. For such maps, we prove the existence of an algebraic (,+)-action on ⁿ for whichF is invariant. Moreover we give some conditions so thatF*([t ₁,...,t _n-1 ]) is the ring of invariants of .Dedicated to all my friends and my family     

Angular limits of holomorphic functions which satisfy an integrability condition

Let (–1,1), let 2/(1–)p<, letp denote the Hölder conjugate ofp, and let be an open arc of the unit circle. It is shown that, iff is a holomorphic function on the unit disc such that: (i) (1–|z|)log⁺|f(z)| isL ^p -integrable on the sector {r:0f has an infinite asymptotic value has -finite (2–(1+)p)-dimensional Hausdorff, measure, thenf has finite angular limits on a subset of of positive linear measure. In fact, a stronger conclusion will be established.     

Arithmetic of the Modular Function j1,8

Since the genus of the modular curve X_1 (8) = _1 (8) * is zero, we find a field generator j _1,8(z) = ₃(2z)/₃(4z) (₃(z) := _n ein ^2z ) such that the function field over X ₁(8) is (j _1,8). We apply this modular function j _1,8 to the construction of some class fields over an imaginary quadratic field K, and compute the minimal polynomial of the singular value of the Hauptmodul N(j _1,8) of (j _1,8).     

Some totally real embeddings of three-manifolds

LetV be a complex hypersurface in an open subset of ³, and letM be a smooth compact real hypersurface inV. Using a theorem of Gromov we prove that there exist small C¹ perturbations ofM in ³ such that is a totally real submanifold of ³. As a consequence we show that certain quotients of the three-sphere admit totally real embeddings into ³. In some special cases including the real projective three-space we find explicit totally real embeddings into ³. Our construction is similar to that of Ahern and Rudin who found a totally real embedding of the three-sphere into ³.Research supported by a fellowship from the Alfred P. Sloan foundation     

Asymptotics with respect to a parameter of solutions of linear functional-differential equations

We study the question of the number of linearly independent solutions of the equationy ⁽ⁿ⁾ (x)+(Fy) (x)+ⁿ y (x)=0,x [0, 1], in which F is a bounded linear operator acting on various normed function spaces. A number of assertions about the asymptotic behavior of these solutions with respect to , tending to infinity are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1460–1469, November, 1990.