Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

Nevanlinna inequality for intersection of divisors

A Nevanlinna-type inequality is proved for holomorphic mapf:C ^m →M and for intersection of sections of a line bundle overM, in which the intersection may not be pure dimensional and the map may be degenerate. Partial financial support was provided by the NSF under grant number DMS-8922760.     

Asplund operators and holomorphic maps

LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0. Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

Branched two-sheeted covers

In this paper we explore the connection between Weierstrass points of subspaces of the holomorphic differentials and the geometry of the canonical curve inPC ^g−1. In particular, we consider non-hyperelliptic Riemann surfaces with involution and the Weierstrass points of the −1 eigenspace of the holomorphic differentials. The case of coverings of a torus is considered in detail. Research of the first author supported in part by the Paul and Gabriella Rosenbaum Foundation, the Landau Center for Research in Mathematical Analysis (supported by Minerva Foundation-Germany) and a US-Israel BSF grant. Research by the second author supported in part by NSF Grant DMS 9003361 and a Lady Davis Visiting Professorship at the Hebrew University.     

Integral representation with weights II, division and interpolation

Let f be a r×m-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic ψ such that ϕ=f ψ, provided that ϕ is holomorphic and annihilates a certain residue current with support on the set where f is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case r=1. The author was partially supported by the Swedish Research Council     

Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture

We show that proving the conjectured sharp constant in a theorem of Dennis Sullivan concerning convex sets in hyperbolic 3-space would imply the Brennan conjecture. We also prove that any conformal mapf:D→Ω can be factored as aK-quasiconformal self-map of the disk (withK independent of Ω) and a mapg:D→Ω with derivative bounded away from zero. In particular, there is always a Lipschitz homeomorphism from any simply connected Ω (with its internal path metric) to the unit disk. The author is partially supported by NSF Grant DMS 9800924.     

On coherent systems of type (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">d</Emphasis>, <Emphasis Type="Italic">n</Emphasis> + 1) on Petri curves

We study coherent systems of type (n, d, n + 1) on a Petri curve X of genus g ≥ 2. We describe the geometry of the moduli space of such coherent systems for large values of the parameter α. We determine the top critical value of α and show that the corresponding “flip” has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of α, proving in many cases that the condition for non-emptiness is the same as for large α. We give some detailed results for g ≤ 5 and applications to higher rank Brill–Noether theory and the stability of kernels of evaluation maps, thus proving Butler’s conjecture in some cases in which it was not previously known. The authors are members of the research group VBAC (Vector Bundles on Algebraic Curves). The first two authors were supported by EPSRC grant GR/T22988/01 for a visit to the University of Liverpool. The second author acknowledges the support of CONACYT grant 48263-F. The third author thanks CIMAT, Guanajuato, México and California State University Channel Islands, where a part of this paper was completed, and acknowledges support from the Academia Mexicana de Ciencias, under its exchange agreement with the Royal Society of London.     

Almost hermitian manifolds and Osserman condition

Let(M, g, J) be an almost Hermitian manifold. In this paper we study holomorphically nonnegatively(δ,Δ)-pinched almost Hermitian manifolds. In [3] it was shown that for such Kahler manifolds a plane with maximal sectional curvature has to be a holomorphic plane(J-invariant). Here we generalize this result to arbitrary almost Hermitian manifolds with respect to the holomorphic curvature tensorH R and toRK-manifolds of a constant type λ(p). In the proof some estimates of the sectional curvature are established. The results obtained are used to characterize almost Hermitian manifolds of constant holomorphic sectional curvature (with respect to holomorphic and Riemannian curvature tensor) in terms of the eigenvalues of the Jacobi-type operators, i.e. to establish partial cases of the Osserman conjecture. Some examples are studied. The first author is partially supported by SFS, Project #04M03.     

Extremal holomorphic curves for defect relations

Drasin’s theorem describing meromorphic functions of finite order with maximal sum of deficiencies is extended to holomorphic curves in projective space. A conjecture about holomorphic curves extremal for Cartan’s defect relation is discussed. Supported by NSF grant DMS-950036. This paper was written at the Norwegian Technology and Science University (NTNU, Trondheim), which the author thanks for its hospitality.     

Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps

We show equivalence of several standard conditions for non-uniform hyperbolicity of complex rational functions, including the Topological Collet-Eckmann condition (TCE), Uniform Hyperbolicity on Periodic orbits, Exponential Shrinking of components of pre-images of small discs, backward Collet-Eckmann condition at one point, positivity of the infimum of Lyapunov exponents of finite invariant measures on the Julia set. The condition TCE is stated in purely topological terms, so we conclude that all these conditions are invariant under topological conjugacy.?For rational maps with one critical point in Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions. Thus the latter ones are invariant by topological conjugacy in the unicritical setting. We also prove that neither part of this stronger statement is valid in the multicritical case. Oblatum 2-IV-2002 & 2-V-2002?Published online: 6 August 2002 RID="*" ID="*"All authors are supported by the European Science Foundation program PRODYN. The first author is also supported by the Foundation for Polish Sciences and Polish KBN grant 2P03A 00917. The second author is grateful to IMPAN and KTH for hospitality and is also supported by a Polish-French governmental agreement, Fundacion Andes and a “Beca Presidente de la Republica,” Chile. The third author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.     

Non-uniform hyperbolicity and universal bounds for S-unimodal maps

An S-unimodal map f is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the infimum of the Lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure. We prove that an S-unimodal map satisfies the Collet-Eckmann condition if and only if it has a uniform hyperbolic structure. The equivalence of several non-uniform hyperbolicity conditions follows. One consequence is that some renormalization of an S-unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the Collet-Eckmann condition is satisfied. The proof uses new universal bounds that hold for any S-unimodal map without periodic attractors. Oblatum 4-VII-1996 & 4-VII-1997     

Identity principles for commuting holomorphic self-maps of the unit disc

Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g agree at the common Wolff point up to a certain order of derivatives (no more than 3 if the Wolff point is on the unit circle), then f≡g.     

On the Bloch Constant for Κ-Quasiconformal Mappings in Several Complex Variables

We study the Bloch constant for Κ-quasiconformal holomorphic mappings of the unit ball B of C ⁿ . The final result we prove in this paper is: If f is a Κ-quasiconformal holomorphic mappig of B into C ⁿ such that det(f′(0)) = 1, then f(B) contains a schlicht ball of radius at least where C _n > 1 is a constant depending on n only, and as n→∞. Received June 24, 1998, Accepted January 14, 1999     

On functional equations satisfied by spinor Euler products for Siegel modular forms of genus 2 with characters

It is proved under certain assumptions that spinor Euler products for Siegel eigen cusp forms with characters with respect to the groups Γ² ₀(q) have holomorphic analytical continuation over the whole complex plane and satisfy a functional equation with two gamma-factors. The author was supported in part by Russian Fund of Fundamental Researches, Grant # 99-01-00099.     

The {\overline{\partial}} operator along the leaves and Guichard’s theorem for a complex simple foliation

In (Ann Sc ENS Sér 3 4:361–380, 1887) Guichard proved that, for any holomorphic function g on ${{\mathbb C}}In (Ann Sc ENS Sér 3 4:361–380, 1887) Guichard proved that, for any holomorphic function g on \mathbb C{{\mathbb C}}, there exists a holomorphic function h (on \mathbb C{{\mathbb C}}) such that h - h °t = g{h - h \circ \tau = g} where τ is the translation by 1 on \mathbb C{{\mathbb C}}. In this note we prove an analogous of this theorem in a more general situation. Precisely, let (M,F){(M,{\mathcal F})} be a complex simple foliation whose leaves are simply connected non compact Riemann surfaces and γ an automorphism of F{{\mathcal F}} which fixes each leaf and acts on it freely and properly. Then, the vector space H_F(M){{\mathcal H}_{\mathcal F}(M)} of leafwise holomorphic functions is not reduced to functions constant on the leaves and for any g ? H_F(M){g \in {\mathcal H}_{\mathcal F}(M)}, there exists h ? H_F(M){h \in {\mathcal H}_{\mathcal F}(M)} such that h - h °g = g{h - h \circ \gamma = g}. From the proof of this theorem we derive a foliated version of Mittag–Leffler Theorem.     

Laplacians on the holomorphic tangent bundle of a Kaehler manifold

Let M be a connected complex manifold endowed with a Hermitian metric g. In this paper, the complex horizontal and vertical Laplacians associated with the induced Hermitian metric 〈·, ·〉 on the holomorphic tangent bundle T ^1,0 M of M are defined, and their explicit expressions are obtained. Using the complex horizontal and vertical Laplacians associated with the Hermitian metric 〈·, ·〉 on T ^1,0 M, we obtain a vanishing theorem of holomorphic horizontal p forms which are compactly supported in T ^1,0 M under the condition that g is a Kaehler metric on M.     

CS Modules Relative to a Torsion Theory

In this article we introduce two new concepts, those of a τ-CS and a s-τ-CS module, which are both torsion-theoretic analogues of CS modules. We investigate their relationship with the familiar concepts of τ-injective, τ-simple and τ-uniform modules and compare them with τ-complemented (τ-injective) modules, which were considered by other authors as torsion-theoretic analogues of CS modules. We are interested in decomposing a relatively CS module into indecomposable submodules, and in determining when a direct sum of relatively CS modules is relatively CS. This paper forms part of the Ph.D. thesis of the first author, written under the supervision of the second author. The first author gratefully acknowledges the support of the Commonwealth Scholarship and Fellowship Committee of New Zealand.     

Monotone Nonincreasing Random Fields on Partially Ordered Sets. II. Probability Distributions on Polyhedral Cones

In this part of the paper, we investigate the structure of an arbitrary measure μ supported by a polyhedral cone C in R ^d in the case where the decumulative distribution function g_μ of the measure μ satisfies certain continuity conditions. If a face Γ of the cone C satisfies appropriate conditions, the restriction μ|_Γint of the measure μ to the interior part of Γ is proved to be absolutely continuous with respect to the Lebesgue measure λ_Γ on the face Γ. Besides, the density of the measure μ|_Γint is expressed as the derivative of the function g_μ multipied by a constant. This result was used in the first part of the paper to find the finite-dimensional distributions of a monotone random field on a poset. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 5–56.     

Construction of constant mean curvature <Emphasis Type="Italic">n</Emphasis>-noids from holomorphic potentials

Using a Weierstrass type representation of constant mean curvature surfaces, we give a general method for constructing constant mean curvature n-noids (of genus 0) from holomorphic potentials, where n ≥ 3. The ends of these surfaces are embedded and asymptotically approach Delaunay surfaces, while the surfaces are in general not even almost embedded. In particular, a 3-parameter family of constant mean curvature trinoids is constructed. Part of this work was done, while the first named author held a Lehrstuhlvertretung at the University of Augsburg. He would like to thank the University of Augsburg for its hospitality. He would also like to acknowledge partial support by DFG-grant DO 776.     

Holomorphie quaternionienne

The paper deals with several aspects of Fueter-holomorphic functions. In the first part a Cauchy-type formula as well as a Morera-type theorem are proved. The second part is concerned with “hemiharmonic” functions which are solutions of δ² f = 0 and are closely related to holomorphic functions. They satisfy a “Mean value” theorem. In the third part new characterizations of holomorphy are given. The fourth part is a study of homogeneous hemiharmonic and holomorphic functions.