The Schwarzian derivative in several complex variables IV

Using two different Lie groups, two different Schwarzian derivatives of holomorphic mappings on domains in Cⁿ are defined and discussed. The necessary and sufficient conditions for annihilation of these two different Schwarzian derivatives are given. Project supported by the National Natural Science Foundation of China.     

多复变数的Schwarz导数(Ⅴ)

本文从Thurston的观点出发,用二阶逼近来定义与讨论矩阵空间C~(m×n)（m≤n）中的域上全纯映照的Schwarz导数及高阶Schwarz导数,证明：如果它们存在的话,那么它们是在R_I（m,n）的紧对偶空间CG（m,n）的全纯自同构群下的相似不变量．并证明：这样得到的Schwarz导数与前几文［1－4］中由Ahlfors的观点得到的Schwarz导数是相一致的．此外,还应用这种观点定义与讨论了C~N中的域上全纯映照的Schwarz导数．     

Applications of the generalized Schwarzian derivative to the analysis of bifurcations of limit cycles

We state and prove a theorem on the equality of the real part of the generalized Schwarzian derivative computed along a bifurcating limit cycle of a family of vector fields defined in ℝ ⁿ to the first Lyapunov quantity of the corresponding Poincaré map.     

Singular complete integrability

We show that a holomorphic vector field in a neighbourhood of its singular point 0∈C ⁿ is analytically normalizable if it has a sufficiently large number of commuting holomorphic vector fields, a sufficiently large number of formal first integrals and that a diophantine small divisors condition related to the linear parts of its centralizer is satisfied.     

Solution to n-dimensional Sturm-Liouville-like equations using multi-dimensional Schwarzian

In the present work we produce the solution to the n-dimensional Sturm-Liouville-like equations in Rn. To make it, we define the multi-dimensional Schwarzian derivative of a real-valued function of n variables and show that its basic properties related to its invariance under the action of a group of multi-dimensional Möbius transformations defined in Rn correspond to a straightforward generalization of those of the one-dimensional Schwarzian.     

A theorem in complex symplectic geometry

We prove that two simple, closed, real-analytic curves in C²ⁿ that are polynomially convex are equivalent under the group of symplectic holomorphic automorphisms of C²ⁿ if and only if the two curves have the same action integral. Every two simple real-analytic arcs in C²ⁿ are so equivalent.     

Differential operators associated with holomorphic mappings

Two differential operators which act on holomorphic mappings to complex projective space are studied. One operator is of second order and characterizes projective linear mappings. The other operator is of third order and may be viewed as a curvature. The two operators together play a role analogous to the Schwarzian derivative.A canonical approximation to a holomorphic mapping is defined, and a relationship between the approximation and the operators is derived. In the one variable case, this reduces to a classical result relating the Schwarzian derivative and the best Möbius approximation to a holomorphic function.     

Regularity of discs attached to a submanifold ofC n

Letp be an analytic disc attached to a generating CR-submanifoldM of C ⁿ . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC ^α holomorphic perturbations ofp alongM is a Banach submanifold of (A^α(D))ⁿ are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC ^α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A ^α (D))ⁿ.     

SOME PROPERTIES OF HOLOMORPHIC CLIFFORDIAN FUNCTIONS IN COMPLEX CLIFFORD ANALYSIS

In this article, we mainly develop the foundation of a new function theory of several complex variables with values in a complex Clifford algebra defined on some subdomains of Cn+1, so-called complex holomorphic Cliffordian functions. We define the complex holomorphic Cliffordian functions, study polynomial and singular solutions of the equation D△mf=0, obtain the integral representation formula for the complex holomorphic Cliffordian functions with values in a complex Clifford algebra defined on some submanifolds of Cn+1, deduce the Taylor expansion and the Laurent expansion for them and prove an invariance under an action of Lie group for them.     

Characterization of the unit ball in C<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> among complex manifolds of dimension<Emphasis Type="BoldItalic">n</Emphasis>

We show that if the group of holomorphic automorphisms of a connected complex manifold M of dimension n is isomorphic as a topological group equipped with the compact-open topology to the automorphism group of the unit ball B ⁿ ⊂ ℂ ⁿ ,then M is biholomorphically equivalent to B ⁿ.     

Globalization of Holomorphic Actions on Principal Bundles

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification G^C of G on. X, this lifts to a holomorphic action of G^C on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification G^C of G acts holomorphically and transitively on G/H. We also show that the ends of a homogeneous complex manifold G/H with more than two ends essentially come from a space of the form S/Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

Closed polylines inscribed in a closed space curve

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in \mathbbRⁿ {\mathbb{R}^n} ,then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.     

The Julia-Wolff-Carathéodory theorem in polydisks

The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls inC ⁿ and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of the main tools for the proof is a general version of the Lindelöf principle valid for not necessarily bounded holomorphic functions.     

Proper holomorphic maps on an irreducible bounded symmetric domain of classical type

In this paper it is extended a result by W. Rudin on proper holomorphic maps from the open ball ofC ⁿ intoC ⁿ. I show that, forn>1, every proper map from an irreducible bounded symmetric domain of classical type intoC ⁿ, can be obtained, module biholomorphic maps, via a finite reflection group.     

Interpolation by holomorphic automorphisms and embeddings in Cn

Let n > 1 and let C ⁿ denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C ⁿ →C ⁿ and for holomorphic automorphisms of C ⁿ on discrete subsets of C ⁿ.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C ⁿ.For each closed complex submanifold (or subvariety) M ⊂ C ⁿ of complex dimension m < n we construct a domain Ω ⊂C ⁿ containing M and a biholomorphic map F: Ω → C ⁿ onto C ⁿ with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C ^n−m →C ⁿ at infinitely many points. If m = n − 1, we construct F as above such that C ⁿ ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C ^m →C ^m−1 such that the complement C ^m+1 ∖F(C ^m )is hyperbolic.     

Absolutely continuous invariant measures forC 2 unimodal maps satisfying the Collet-Eckmann conditions

Summary In this paper we show that unimodal mappingsf[0, 1][0, 1] have absolutely continuous measures of positive entropy if these maps areC ² and satisfy the so-called Collet-Eckmann conditions. No conditions on the Schwarzian derivative off are assumed.     

On proper holomorphic self-maps of generalized complex ellipsoids

The purpose of this paper is to prove that every proper holomorphic self-mapping of a Reinhardt domain Ω in C ⁿ which is a generalization of a complex ellipsoid is biholomorphic. The main novelty of our result is that Ω is a domain in C ⁿ such that it is allowed to have a boundary point at which the Levi determinant has infinite order of vanishing.     

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that homologically nontrivial generic smooth (2n−1)-parameter families of analytic discs in Cn, n?2, attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C² as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C² in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik-Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in Cn extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2).     

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