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.     

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.     

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.     

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.     

Invariant subrings of ℂ[X,Y,Z] which are complete intersections

It is known that for every finite subgroup G of SL(2,), the invariant subring [X,Y]^G is a hyper-surface. In this note we treat finite subgroups of SL(3,) and give complete classification of the finite subgroups of SL(3,) whose invariant subrings are complete intersections.     

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     

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     

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.     

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.     

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     

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 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.     

Asymptotic behavior of the Jost function of a two-dimensional Schrödinger operator

We find the asymptotic behavior of the Jost function(Z,) of a two-dimensional Schrödinger operator for arbitrary and Z/|Z|S¹ as |Z| We discuss consequences of the asymptotic formulas for the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 96—103, 1988.     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.     

On the Monodromy of a Multivalued Function along Its Ramification Locus

We consider multivalued analytic functions in ⁿ) whose set of singular points occupies a very small part of ⁿ). Under a mapping of a topological space Y into ⁿ), such a function f can induce a multivalued function on Y. This is possible even if the image of Y entirely lies in the ramification set of f. We estimate the monodromy group of the induced function via the monodromy group of f.     

Nicht endlich erzeugte Primideale in Steinschen Algebren

In this note we exhibit a closed prime idealF in the ring Ó(³) of all holomorphic functions on ³ which is not finitely generated.F is the ideal of a certain irreducible curve Y³, obtained as the image of a proper holomorphic map f³.

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Herrn Karl Stein gewidmet     

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.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

On four-sheeted polynomial mappings of ℂ2. I. The case of an irreducible ramification curve

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf²² with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 847–862, December, 1998.The authors are grateful to A. G. Vitushkin and P. Cassou-Nogues for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01218. The work of the second author was done under the financial support of DGICYT (Spain), grant No. SAB95-0502.     

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