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.     

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     

Ordnungseigenschaften holomorpher Funktionen

A linear geometric order theory for holomorphic mappings F: ^nm is given if a family of linear subspaces L of ⁿ×^m is specified; the theory then is concerned with the order number of connected parts of the intersections L (F) of the L's with the graph (F) of F. For m=1 (i.e. case of one function) it is proved that the local linear order number of a holomorphic function is finite in every point. If the L's are real-linear subspaces then real differential geometric methods lead to the proof, if the L's are complex-linear then ideal-theoretical means do.     

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     

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.     

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.     

Quasisimilarity of model contractions with unequal defects

Let T and T be C₁₀ contractions with characteristic functions H (ⁿⁿ⁺¹), H (^mm+1). The fundamental result is: T and T are quasisimilar if and only if The paper contains an analysis of this condition; examples are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 24–37, 1986.     

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ², i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over ² are holomorphic. This classification provides a one-parameter family of examples of Lagrangian spheres in ².Research partially supported by a DGICYT grant No. PB91-0731.     

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.     

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     

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.     

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     

The shape invariant of triangles and trigonometry in two-point homogeneous spaces

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces P ⁿ and P ⁿ (n2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces H ⁿ and H ⁿ (n2).     

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.     

Willmore Surfaces of ?4 and the Whitney Sphere

We make a contribution to the study of Willmore surfaces infour-dimensional Euclidean space ⁴ by making useof the identification of ⁴ with two-dimensionalcomplex Euclidean space ². We prove that theWhitney sphere is the only Willmore Lagrangian surface of genus zero in⁴ and establish some existence and uniquenessresults about Willmore Lagrangian tori in ^{4 2}.     

A perturbation result for the asymptotic behavior of matrix powers

We prove a perturbation result for the asymptotic behavior of the sequence (A ⁿ c) _nN , whereAG|(d), the space of invertibled×d matrices, andc ^d .     

Harmonic morphisms from complex projective spaces

In this paper we study harmonic morphisms :U P ^m N ² from open subsets of complex projective spaces to Riemann surfaces. We construct many new examples of such maps which are not holomorphic with respect to the standard Kähler structure on P ^m.The research leading to this paper was supported by the Icelandic Science Fund and the Danish National Science Fund.     

Equilibrium triangulations of the complex projective plane

Starting with the well-known 7-vertex triangulation of the ordinary torus, we construct a 10-vertex triangulation of P² which fits the equilibrium decomposition of P² in the simplest possible way. By suitable positioning of the vertices, the full automorphism group of order 42 is realized by a discrete group of isometries in the Fubini-Study metric. A slight subdivision leads to an elementary proof of the theorem of Kuiper-Massey which says that P² modulo conjugation is PL homeomorphic to the standard 4-sphere. The branch locus of this identification is a 7-vertex triangulation P ₂ ⁷ of the real projective plane. We also determine all tight simplicial embeddings of P ₂ ¹⁰ and P ₂ ⁷ .     

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.     

Die Zahl der Spitzen und die Jacobi-Algebra einer isolierten Hyperflächensingularität

Let f{x_o,...,x_n} define a germ of a complex analytic hypersurface (X_o,0) with isolated singularity. We show that the number of cusps of the unfolded discriminant curve is an invariant of the Jacobian algebra {x,_o},...,x_n/(f/x_o,...,f/x_n) of f. Moreover we show that this number + 1 equals the sum of the Milnor numbers of (X_o,0) and of the polar curve of (X_o,0). Our result generalizes formulas of Iversen and Lê for plane curves to arbitrary dimensions.