Bouquet and join theorems for disentanglements

The disentanglement of certain augmentations is shown to be the topological join of a disentanglement and a Milnor fibre. The kth disentanglement of a finite map is defined and for corank 1 maps from ℂ ⁿ to ℂ ^{n +1} it is shown that they are homotopically equivalent to a wedge of spheres. Applications to the Mond conjecture are given. Oblatum 24-VII-2000 & 5-VII-2001?Published online: 12 October 2001     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP ⁿ . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP ⁿ and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same kind of investigation also for the complex hyperbolic space ℂH ⁿ . Most of the results are completely analogous, however, there are also some interesting differences. Received: 15 January 1996     

On weighted spaces of holomorphic functions of several variables

We generalize the results of [11] and [12] for the unit ball $ \mathbb{B}_d $ \mathbb{B}_d of ℂ ^d . In particular, we show that under the weight condition (B) the weighted H ^∞-space on $ \mathbb{B}_d $ \mathbb{B}_d is isomorphic to ℓ^∞ and thus complemented in the corresponding weighted L ^∞-space. We construct concrete, generalized Bergman projections accordingly. We also consider the case where the domain is the entire space ℂ ^d . In addition, we show that for the polydisc $ \mathbb{D}^d $ \mathbb{D}^d ^d , the weighted H ^∞-space is never isomorphic to ℓ^∞.     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

On the roper-suffridge extension operator

In 1995, Roper and Suffridge defined an extension operator which maps a locally biholomorphic function on the unit diskD in ℂ to a locally biholomorphic mapping on the unit ballB ⁿ in ℂⁿ. This extension operator preserves many important properties, e.g., convexity and starlikeness, etc. In this note, we introduce the family ofε starlike mappings, and prove that the Roper-Suffridge extension operator preserves theε starlikeness on some Reinhardt domains. This result includes many known results and solves an open problem of Graham and Kohr. Project supported by the National Science Foundation of China.     

The zeta function of a simplicial complex

Given a simplicial complex δ on vertices {1, …,n} and a fieldF we consider the subvariety of projective (n−1)-space overF consisting of points whose homogeneous coordinates have support in δ. We give a simple rational expression for the zeta function of this singular projective variety overF _q and show a close connection with the Betti numbers of the corresponding variety over ℂ. This connection is particularly simple in the case when Δ is Cohen-Macaulay.     

Interaction of Legendre curves and Lagrangian submanifolds

It is proved in [8] that there exist no totally umbilical Lagrangian submanifolds in a complex-space-form ,n≥2, except the totally geodesic ones. In this paper we introduce the notion of LagrangianH-umbilical submanifolds which are the “simplest” Lagrangian submanifolds next to the totally geodesic ones in complex-space-forms. We show that for each Legendre curve in a 3-sphereS ³ (respectively, in a 3-dimensional antide Sitter space-timeH ₁ ³ ), there associates a LagrangianH-umbilical submanifold in ℂP ⁿ (respectively, in ℂH ⁿ ) via warped products. The main part of this paper is devoted to the classification of LagrangianH-umbilical submanifolds in ℂP ⁿ and in ℂH ⁿ . Our classification theorems imply in particular that “except some exceptional classes”, LagrangianH-umbilical submanifolds of ℂP ⁿ and of ℂH ⁿ are obtained from Legendre curves inS ³ or inH ₁ ³ via warped products. This provides us an interesting interaction of Legendre curves and LagrangianH-umbilical submanifolds in non-flat complex-space-forms. As an immediate by-product, our results provide us many concrete examples of LagrangianH-umbilical isometric immersions of real-space-forms into non-flat complex-space-forms.     

Vanishing Topology of Codimension 1 Multi-germs over ℝ and ℂ

We construct all A _e -codimension 1 multi-germs of analytic (or smooth) maps (k ⁿ , T) (k ^p , 0), with n p – 1, (n, p) nice dimensions, k = or , by augmentation and concatenation operations, starting from mono-germs (|T| = 1) and one 0-dimensional bi-germ. As an application, we prove general statements for multi-germs of corank 1: every one has a real form with real perturbation carrying the vanishing homology of the complexification, every one is quasihomogeneous, and when n = p – 1 every one has image Milnor number equal to 1 (this last is already known when n p).     

Determination of the poles of the topological zeta function for curves

Tof ∈ℂ[x ₁…,x _n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f ⁻¹{0} inf ⁻¹{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x ₁,x ₂]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution. The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research N.F.W.O.     

Rational holomorphic maps from \mathbbB2 \mathbb{B}^2 into \mathbbBN \mathbb{B}^N with degree 2

Rational proper holomorphic maps from the unit ball in ℂ² into the unit ball ℂ ^N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

New examples of Lagrangian rigidity

Letn=4 or 8. We prove that any Lagrangian embedding ofS ^{n − 1} ×S ¹ into ℂ ⁿ has a trivial linking class. We deduce that every embedding ofS ³ ×S ⁴ into ℂ⁴ is isotopic to a Lagrangian embedding. This is false ifn = 8.     

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

Minimal dimension families of complex lines sufficient for holomorphic extension of functions

We consider continuous functions given on the boundary of a bounded domain D in ℂ ⁿ , n > 1, with the one-dimensional holomorphic extension property along families of complex lines. We study the existence of holomorphic extensions of these functions to D depending on the dimension and location of the families of complex lines.     

On the Equicontinuity Region of Discrete Subgroups of <Emphasis Type="Italic">PU</Emphasis>(1,<Emphasis Type="Italic">n</Emphasis>)

Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ_ℂ ⁿ preserving the unit ball ℍ_ℂ ⁿ , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ_ℂ ⁿ and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ_ℂ ⁿ which are tangent to ∂ℍ_ℂ ⁿ at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍ_ℂ ⁿ which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.     

Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ _ℂ ⁿ . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ _ℂ ³ for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.     

The element number of the convex regular polytopes

Let Σ be a set of n-dimensional polytopes. A set Ω of n-dimensional polytopes is said to be an element set for Σ if each polytope in Σ is the union of a finite number of polytopes in Ω identified along (n − 1)-dimensional faces. In this paper, we consider the n-dimensional polytopes in general, and extend the notion of element sets to higher dimensions. In particular, we will show that in the 4-space, the element number of the six convex regular polychora is at least 2, and in the n-space (n ≥ 5), the element number is 3, unless n + 1 is a square number.