Unnamed: 0
int64 0
41k
| title
stringlengths 4
274
| category
stringlengths 5
18
| summary
stringlengths 22
3.66k
| theme
stringclasses 8
values |
|---|---|---|---|---|
700 |
On the Removable Singularities for Meromorphic Mappings
|
math.CV
|
If E is a nonempty closed subset of the locally finite Hausdorff
(2n-2)-measure on an n-dimensional complex manifold M and all points of E are
nonremovable for a meromorphic mapping of M \ E into a compact K\"ahler
manifold, then E is a pure (n-1)-dimensional complex analytic subset of M.
|
math
|
701 |
Szegö kernels for certain unbounded domains in $\Bbb C^2$
|
math.CV
|
No abstract available.
|
math
|
702 |
Domains in $\cx {n+1}$ with Noncompact Automorphism Group. II
|
math.CV
|
No abstract available.
|
math
|
703 |
Zero sets of some classes of entire functions
|
math.CV
|
A method of constructing an entire function with given zeros and estimates of
growth is suggested. It gives a possibility to describe zero sets of certain
classes of entire functions of one and several variables in terms of growth of
volume of these sets in certain polycylinders.
|
math
|
704 |
Analytic varieties versus integral varieties of Lie algebras of vector fields
|
math.CV
|
We associate to any germ of an analytic variety a Lie algebra of tangent
vector fields, the {\it tangent algebra}. Conversely, to any Lie algebra of
vector fields an analytic germ can be associated, the {\it integral variety}.
The paper investigates properties of this correspondence: The set of all
tangent algebras is characterized in purely Lie algebra theoretic terms. And it
is shown that the tangent algebra determines the analytic type of the variety.
|
math
|
705 |
Holomorphic curvature of Finsler metrics and complex geodesics
|
math.CV
|
In his famous 1981 paper, Lempert proved that given a point in a strongly
convex domain the complex geodesics (i.e., the extremal disks) for the
Kobayashi metric passing through that point provide a very useful fibration of
the domain. In this paper we address the question whether, given a smooth
complex Finsler metric on a complex manifold, it is possible to give purely
differential geometric properties of the metric ensuring the existence of such
a fibration in complex geodesics of the manifold. We first discuss at some
length the notion of holomorphic sectional curvature for a complex Finsler
metric; then, using the differential equation of complex geodesics we obtained
in a previous paper, we show that for every pair (point, tangent vector) there
is a (only a segment if the metric is not complete) complex geodesic passing
through the point tangent to the given vector iff the Finsler metric is
K\"ahler, has constant holomorphic sectional curvature -4 and satisfies a
simmetry condition on the curvature tensor. Finally, we show that a complex
Finsler metric of constant holomorphic sectional curvature -4 satisfying the
given simmetry condition on the curvature is necessarily the Kobayashi metric.
|
math
|
706 |
Sequences of analytic disks
|
math.CV
|
The subject considered in this paper has, at least, three points of interest.
Suppose that we have a sequence of one-dimensional analytic varieties in a
domain in $\Bbb C^n$. The cluster of this sequence consists from all points in
the domains such that every neighbourhood of such points intersects with
infinitely many different varieties. The first question is: what analytic
properties does the cluster inherit from varieties? We give a sufficient
criterion when the cluster contains an analytic disk, but it follows from
examples of Stolzenberg and Wermer that, in general, clusters can contain no
analytic disks. So we study algebras of continuous function on clusters, which
can be approximated by holomorphic functions or polynomials, and show that this
algebras possess some analytic properties in all but explicitly pathological
and uninteresting cases. Secondly, we apply and results about clusters to
polynomial hulls and maximal functions, finding remnants of analytic structures
there too. And, finally, due to more and more frequent appearances of analytic
disks as tools in complex analysis, it seems to be interesting to look at their
sequences to establish terminology, basic notation and properties.
|
math
|
707 |
A counterexample to the Arakelyan Conjecture
|
math.CV
|
A ``self--similar'' example is constructed that shows that a conjecture of N.
U. Arakelyan on the order of decrease of deficiencies of an entire function of
finite order is not true.
|
math
|
708 |
The Green function of Teichmüller spaces with applications
|
math.CV
|
We describe briefly a new approach to some problems related to Teichm\"uller
spaces, invariant metrics, and extremal quasiconformal maps. This approach is
based on the properties of plurisubharmonic functions, especially of the
plurisubharmonic Green function. The main theorem gives an explicit
representation of the Green function for Teichm\"uller spaces by the
Kobayashi-Teichm\"uller metric of these spaces. This leads to various
applications. In particular, this gives a new characterization of extremal
quasiconformal maps.
|
math
|
709 |
Radó theorem and its generalization for CR-mappings
|
math.CV
|
The following theorem is proved:
Let M be a locally Lipschitz hypersurface in C^n with one-sided extension
property at each point (e.g., without analytic discs). Let S be a closed subset
of M and f : M \ S ---> C^m \ E is a CR-mapping of class L^{\infty} such that
the cluster set of f on S along of Lebesque points of f is contained in a
closed complete pluripolar set E. Then there is a CR-mapping \~f : M ---> C^m
of class L^{\infty}(M) such that \~f |M\S = f. It follows also that S is
removable for CR \cap L^{\infty} (M \ S).
|
math
|
710 |
Complexity of the classical kernel functions of potential theory
|
math.CV
|
We show that the Bergman, Szego, and Poisson kernels associated to a finitely
connected domain in the plane are all composed of finitely many easily computed
functions of one variable. The new formulas give rise to new methods for
computing the Bergman and Szeg\H o kernels in which all integrals used in the
computations are line integrals; at no point is an integral with respect to
area measure required. The results mentioned so far can be interpreted as
saying that the kernel functions are simpler than one might expect. However, we
also prove that the kernels cannot be too simple by showing that the only
finitely connected domains in the plane whose Bergman or Szeg\H o kernels are
rational functions are the obvious ones. This leads to a proof that the
classical Green's function associated to a finitely connected domain in the
plane is the logarithm of a rational function if and only if the domain is
simply connected and rationally equivalent to the unit disc.
|
math
|
711 |
Moduli of bounded holomorphic functions in the ball
|
math.CV
|
We prove that there is a continuous non-negative function $g$ on the unit
sphere in $\cd$, $d \geq 2$, whose logarithm is integrable with respect to
Lebesgue measure, and which vanishes at only one point, but such that no
non-zero bounded analytic function $m$ in the unit ball, with boundary values
$m^\star$, has $|m^\star| \leq g$ almost everywhere. The proof analyzes the
common range of co-analytic Toeplitz operators in the Hardy space of the ball.
|
math
|
712 |
Complex Finsler metrics
|
math.CV
|
In this paper we describe an approach to complex Finsler metrics suitable to
deal with global questions, and stressing the similarities between hermitian
and complex Finsler metrics. Let $F$ be a smooth complex Finsler metric on a
complex manifold $M$, and assume that the indicatrices of $F$ are strongly
pseudoconvex -- we shall say that $F$ itself is strongly pseudoconvex.
The vertical bundle $\cal V$ is the kernel of the differential of the
canonical projection of the holomorphic tangent bundle of $M$. Using $F$, it is
possible to endow $\cal V$ with a hermitian metric; let $D$ be the Chern
connection associated to this metric. It turns out that there is a canonical
way to build starting from $D$ a horizontal bundle $\cal H$, as well as a
bundle isomorphism $\Theta\colon{\cal V}\to\cal H$. Using $\Theta$ we may
transfer both the metric and the connection on $\cal H$; furthermore, there is
a canonical isometric embedding $\chi$ of the holomorphic tangent bundle of $M$
into $\cal H$.
Our idea is that the Finsler geometry of $M$ can be studied applying standard
hermitian techniques to $\cal H$ using $\chi$ to transfer back and forth
problems and solutions. To support this claim, in this paper we discuss Bianchi
identities, K\"ahler conditions, the first and second variation formulas,
geodesics and holomorphic curvature. Furthermore, we provide a sound geometric
interpretation to our previous work on the existence of complex geodesic
curves. Finally, we prove that in complex K\"ahler Finsler manifolds with
constant nonpositive holomorphic curvature (and satisfying an additional
symmetry property on the curvature) the complex geodesic curves define a nice
fibration of the manifold, completely analogous to the one described by Lempert
in strongly convex domains.
|
math
|
713 |
Quasiconformal Homeomorphisms on CR 3-Manifolds With Symmetries
|
math.CV
|
An extremal quasiconformal homeomorphisms in a class of homeomorphisms
between two CR 3-manifolds is an one which has the least conformal distortion
among this class. This paper studies extremal quasiconformal homeomorphisms
between CR 3-manifolds which admit transversal CR circle actions. Equivariant
$K$-quasiconformal homeomorphisms are characterized by an area-preserving
property and the $K$-quasiconformality of their quotient maps on the spaces of
$S^1$-orbits. A large family of invariant CR structures on $S^3$ is constructed
so that the extremal quasiconformal homeomorphisms among the equivariant
mappings between them and the standard structure are completely determined.
These homeomorphisms also serve as examples showing that the extremal
quasiconformal homeomorphisms between two invariant CR manifolds are not
necessarily equivariant.
|
math
|
714 |
Regularity And Extremality Of Quasiconformal Homeomorphisms On CR 3-Manifolds
|
math.CV
|
This paper first studies the regularity of conformal homeomorphisms on smooth
locally embeddable strongly pseudoconvex CR manifolds. Then moduli of curve
families are used to estimate the maximal dilatations of quasiconformal
homeomorphisms. On certain CR 3-manifolds, namely, CR circle bundles over flat
tori, extremal quasiconformal homeomorphisms in some homotopy classes are
constructed. These extremal mappings have similar behaviors to Teichm\"uller
mappings on Riemann surfaces.
|
math
|
715 |
Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces
|
math.CV
|
We will announce two theorems. The first theorem will classify all
topological types of degenerate fibers appearing in one-parameter families of
Riemann surfaces, in terms of ``pseudoperiodic'' surface homeomorphisms. The
second theorem will give a complete set of conjugacy invariants for the mapping
classes of such homeomorphisms. This latter result implies that Nielsen's set
of invariants [{\it Surface transformation classes of algebraically finite
type}, Collected Papers 2, Birkh\"auser (1986)] is not complete.
|
math
|
716 |
Cartwright-type and Bernstein-type theorems for functions analytic in a cone
|
math.CV
|
Cartwright-type and Bernstein-type theorems, previously known only for
functions of exponential type in $\C^n$, are extended to the case of functions
of arbitrary order in a cone.
|
math
|
717 |
Entire periodic functions with plane zeros
|
math.CV
|
We give a complete description of divisors of entire periodic functions in
$\C^n$ with plane zeros.
|
math
|
718 |
CR manifolds with noncompact connected automorphism groups
|
math.CV
|
The main result of this paper is that the identity component of the
automorphism group of a compact, connected, strictly pseudoconvex CR manifold
is compact unless the manifold is CR equivalent to the standard sphere. In
dimensions greater than 3, it has been pointed out by D. Burns that this result
follows from known results on biholomorphism groups of complex manifolds with
boundary and the fact that any such CR manifold M can be realized as the
boundary of an analytic variety. When M is 3-dimensional, Burns's proof breaks
down because abstract CR 3-manifolds are generically not realizable as
boundaries. This paper provides an intrinsic proof of compactness that works in
any dimension.
|
math
|
719 |
Continuity of the complex Monge-Ampere operator
|
math.CV
|
The complex Monge-Amp\`ere operator $(dd^c)^n$ is an important tool in
complex analysis. It would be interesting to find the right notion of
convergence $u_j\to u$ such that $(dd^cu_j)^n\to (dd^cu)^n$ in the weak
topology. In this paper, using the $C_{n-1}$-capacity, we give a sufficient
condition of the weak convergence $(dd^cu_j)^n\to (dd^cu)^n$. We also show that
our condition is quite sharp in some case.
|
math
|
720 |
The Kobayashi metric for non-convex complex ellipsoids
|
math.CV
|
In the paper we give some necessary conditions for a mapping to be a
$\kappa$-geodesic in non-convex complex ellipsoids. Using these results we
calculate explicitly the Kobayashi metric in the ellipsoids
$\{|z_1|^2+|z_2|^{2m}<1\}\subset\bold C^2$, where $m<\frac12$.
|
math
|
721 |
Propagation of Gevrey Regularity for a Class of Hypoelliptic Equations
|
math.CV
|
We prove results on the propagation of Gevrey and analytic wave front sets
for a class of $C^\infty$ hypoelliptic equations with double characteristics.
|
math
|
722 |
Global (and Local) Analyticity for Second Order Operators Constructed from Rigid Vector Fields on Products of Tori
|
math.CV
|
We prove global analytic hypoellipticity on a product of tori for partial
differential operators which are constructed as rigid (variable coefficient)
quadratic polynomials in real vector fields satisfying the H\"ormander
condition and where $P$ satisfies a `maximal' estimate. We also prove an
analyticity result that is local in some variables and global in others for
operators whose prototype is
$$ P= \left({\partial \over {\partial x_1}}\right)^2 + \left({\partial \over
{\partial x_2}}\right)^2 + \left(a(x_1,x_2){\partial \over {\partial
t}}\right)^2.$$
(with analytic $a(x), a(0)=0,$ naturally, but not identically zero). The
results, because of the flexibility of the methods, generalize recent work of
Cordaro and Himonas in \cite{Cordaro-Himonas 1994} and Himonas in \cite{Himonas
199X} which showed that certain operators known not to be locally analytic
hypoelliptic (those of Baouendi and Goulaouic \cite{Baouendi-Goulaouic 1971},
Hanges and Himonas \cite{Hanges-Himonas 1991}, and Christ \cite{Christ 1991a})
were {\it globally} analytic hypoelliptic on products of tori.
|
math
|
723 |
The Lu Qi-Keng Conjecture Fails Generically
|
math.CV
|
The bounded domains of holomorphy in~$\mathbf{C}^n$ whose Bergman kernel
functions are zero-free form a nowhere dense subset (with respect to a variant
of the Hausdorff distance) of all bounded domains of holomorphy.
|
math
|
724 |
On the reflection principle in C^n
|
math.CV
|
We propose a reflection principle for holomorphic objects in ${\Bbb C}^n$.
Our construction generalizes the classical principle of H.Lewy, S.Pinchuk and
S.Webster.
|
math
|
725 |
Unbounded Symmetric Homogeneous Domains in Spaces of Operators
|
math.CV
|
We define the domain of a linear fractional transformation in a space of
operators and show that both the affine automorphisms and the compositions of
symmetries act transitively on these domains. Further, we show that Liouville's
theorem holds for domains of linear fractional transformations, and, with an
additional trace class condition, so does the Riemann removable singularities
theorem. We also show that every biholomorphic mapping of the operator domain
$I < Z^*Z$ is a linear isometry when the space of operators is a complex Jordan
subalgebra of ${\cal L}(H)$ with the removable singularity property and that
every biholomorphic mapping of the operator domain $I + Z_1^*Z_1 < Z_2^*Z_2$ is
a linear map obtained by multiplication on the left and right by J-unitary and
unitary operators, respectively.
Readers interested only in the finite dimensional case may identify our
spaces of operators with spaces of square and rectangular matrices.
|
math
|
726 |
On extremal mappings in complex ellipsoids
|
math.CV
|
In the paper we generalize the notion of problem (P) introduced by Poletsky.
We introduce the notion of (P_m) extremals. For example, geodesics are (P_1)
extremals. Using obtained results we present a description of (P_m) extremals
in arbitrary complex ellipsoids. It is a generalization of the result obtained
by Jarnicki-Pflug-Zeinstra. We also have a proof of conjecture put forward by
Pflug-Zwonek concerning the formulas for geodesics in non-convex complex
ellipsoids.
|
math
|
727 |
Shulim Kaliman and Mikhail Zaidenberg
|
math.CV
|
No abstract available.
|
math
|
728 |
Regularity of CR mappings between algebraic hypersurfaces
|
math.CV
|
We prove that if $M$ and $M'$ are algebraic hypersurfaces in $ C^ N$, i.e.
both defined by the vanishing of real polynomials, then any sufficiently smooth
CR mapping with Jacobian not identically zero extends holomorphically provided
the hypersurfaces are holomorphically nondegenerate . Conversely, we prove that
holomorphic nondegeneracy is necessary for this property of CR mappings to
hold. For the case of unequal dimensions, we also prove that if $M$ is an
algebraic hypersurface in $ C^N$ which does not contain any complex variety of
positive codimension and $M'$ is the sphere in $ C^{N+1 }$ , then extendability
holds for all CR mappings with certain minimal a priori regularity.
Theorem A. Let $M$ and $M'$ be two algebraic hypersurfaces in $C^N$ and
assume that $M$ is connected and holomorphically nondegenerate. If $H$ is a
smooth CR mapping from $M$ to $M'$ with $ Jac H \not\equiv 0$, where $Jac H$ is
the Jacobian determinant of $H$, then $H$ extends holomorphically in an open
neighborhood of $M$ in $ C^N$.
A recent example given by Ebenfelt shows that the conclusion of Theorem A
need not hold if $M$ is real analytic, but not algebraic.
Theorem B. Let $M$ be a connected real analytic hypersurface in $C^N$ which
is holomorphically degenerate at some point $p_1$. Let $p_0 \in M$ and suppose
there exists a germ at $p_0$ of a smooth CR function on $M$ which does not
extend holomorphically to any full neighborhood of $p_0$ in $C^N$. Then there
exists a germ at $p_0$ of a smooth CR diffeomorphism from $M$ into itself,
fixing $p_0$, which does not extend holomorphically to any neighborhood of
$p_0$ in $C^N$.
Theorem C. Let $M\subset C^N$ be an algebraic hypersurface. Assume that there
is no nontrivial complex analytic variety contained in $M$ through $p_0 \in M$,
and let $m=m_{p_0}$ be the D'Angelo type. If $H: M \to S^{2N+1}\subset C^{N+1}$
is a CR map of class $C^m$, where $S^{2N+1}$ denotes the boundary of the unit
ball in $C^{N+1 }$, then $H$ admits a holomorphic extension in a neighborhood
of $p_0$.
|
math
|
729 |
Fixed points of elliptic reversible transformations with integrals
|
math.CV
|
We show that for a certain family of integrable reversible transformations,
the curves of periodic points of a general transformation cross the level
curves of its integrals. This leads to the divergence of the normal form for a
general reversible transformation with integrals. We also study the integrable
holomorphic reversible transformations coming from real analytic surfaces in
C^2 with non-degenerate complex tangents. We show the existence of real
analytic surfaces with hyperbolic complex tangents, which are contained in a
real hyperplane, but cannot be transformed into the Moser-Webster normal form
through any holomorphic transformation.
|
math
|
730 |
Divergence of the normalization for real Lagrangian surfaces near complex tangents
|
math.CV
|
We study real Lagrangian analytic surfaces in C^2 with a non-degenerate
complex tangent. Webster proved that all such surfaces can be transformed into
a quadratic surface by formal symplectic transformations of C^2. We show that
there is a certain dense set of real Lagrangian surfaces which cannot be
transformed into the quadratic surface by any holomorphic (convergent)
transformation of C^2. The divergence is contributed by the parabolic character
of a pair of involutions generated by the real Lagrangian surfaces.
|
math
|
731 |
Integrable analytic vector fields with a nilpotent linear part
|
math.CV
|
We study the normalization of integrable analytic vector fields with a
nilpotent linear part. We prove that such an analytic vector field can be
transformed into a certain form by convergent transformations when it has a
non-singular formal integral. In particular, we show that a formally
linearizable analytic vector field with a nilpotent linear part is linearizable
by convergent transformations. We then prove that there are smoothly
linearizable parabolic analytic transformations which cannot be embedded into
the flow of any analytic vector field with a nilpotent linear part.
|
math
|
732 |
Unimodular invariants of totally real tori in C^n
|
math.CV
|
We study the global invariants of real analytic manifolds in the complex
space with respect to the group of holomorphic unimodular transformations. We
consider only totally real manifolds which admits a certain fibration over the
circle. We find a complete set of invariants for totally real tori in C^n which
are close to the standard torus. The invariants are obtained by an analogous
classification of complex-valued analytic $n$-forms on the standard torus. We
also study the realization of certain exact complex-valued analytic $n$-forms
on the standard torus through non-critical totally real embeddings.
|
math
|
733 |
Circle Packings in the Unit Disc
|
math.CV
|
A Bl-packing is a (branched) circle packing that ``properly covers'' the unit
disc. We establish some fundamental properties of such packings. We give
necessary and sufficient conditions for their existence, prove their
uniqueness, and show that their underlying surfaces, known as carriers, are
quasiconformally equivalent to surfaces of classical Blaschke products. We also
extend our earlier approximation results of to general combinatorial patterns
of tangencies in Bl-packings. Finally, a branched version of the Discrete
Uniformization Theorem of Beardon and Stephenson is given.
|
math
|
734 |
A characterization of the finite multiplicity of a CR mapping
|
math.CV
|
No abstract available.
|
math
|
735 |
The Several Complex Variables Problem List
|
math.CV
|
The purpose of this bulletin board is to collect problems in higher
dimensional complex analysis. We are interested both in basic research
questions as well as interactive questions with other fields and sciences. We
encourage everybody to submit problems to the list. This includes not only
those coming up in your own work, but also others- maybe well known and
classical- that you see missing, but that you think workers in the field should
be aware of. Not only are we searching for basic research type questions in
several complex variables, we also solicit questions exploring relations to
other mathematical fields, one complex variable, partial differential
equations, differential geometry, dynamics, etc. and to other sciences such as
physics, engineering, biology etc. While some questions fall rather naturally
into one of the subject areas in this problem list and may lead to a
publishable paper, other questions may be non- specific or of a transient or
technical quality. Thus we have a section called ``Scratchpad'' for
conversational questions, vaguely formulated questions, or questions to which
you may hope to get a quick answer. In the ``open prize problems'' section, you
are welcome to offer a nice little prize to whoever does it. The miscellaneous
section is for announcements of conferences, jokes, remarks on the general
state of the field, etc.
|
math
|
736 |
On $\bold N$-circled $\bold{H^\infty}$-domains of holomorphy
|
math.CV
|
We present various characterizations of $n$-circled domains of holomorphy
$G\subset\CC^n$ with respect to some subspaces of $\Cal H^\infty(G)$.
|
math
|
737 |
Algebraicity of holomorphic mappings between real algebraic sets in ${\bold C}^n$
|
math.CV
|
We give conditions under which a germ of a holomorphic mapping in $\Bbb C^N$,
mapping an irreducible real algebraic set into another of the same dimension,
is actually algebraic.
Let $A\subset \bC^N$ be an irreducible real algebraic set. Assume that there
exists $\po \in A$ such that $A$ is a minimal, generic, holomorphically
nondegenerate submanifold at $\po$. We show here that if $H$ is a germ at $p_1
\in A$ of a holomorphic mapping from $\bC^N$ into itself, with Jacobian $H$ not
identically $0$, and $H(A)$ contained in a real algebraic set of the same
dimension as $A$, then $H$ must extend to all of $\bC^N$ (minus a complex
algebraic set) as an algebraic mapping. Conversely, we show that for any
``model case'' (i.e., $A$ given by quasi-homogeneous real polynomials), the
conditions on $A$ are actually necessary for the conclusion to hold.
|
math
|
738 |
Reinhardt Domains with Non-Compact Automorphism Groups
|
math.CV
|
We give an explicit description of smoothly bounded Reinhardt domains with
noncompact automorphism groups. In particular, this description confirms a
special case of a conjecture of Greene/Krantz.
|
math
|
739 |
Finite Type Conditions on Reinhardt Domains
|
math.CV
|
In this paper we prove that, if $p$ is a boundary point of a smoothly bounded
pseudoconvex Reinhardt domain in $\C^n$, then the variety type at $p$ is
identical to the regular type.
|
math
|
740 |
A stabilization theorem for Hermitian forms and applications to holomorphic mappings
|
math.CV
|
We consider positivity conditions both for real-valued functions of several
complex variables and for Hermitian forms. We prove a stabilization theorem
relating these two notions, and give some applications to proper mappings
between balls in different dimensions. The technique of proof relies on the
simple expression for the Bergman kernel function for the unit ball and
elementary facts about Hilbert spaces. Our main result generalizes to Hermitian
forms a theorem proved by Polya [HLP] for homogeneous real polynomials, which
was obtained in conjunction with Hilbert's seventeenth problem. See [H] and [R]
for generalizations of Polya's theorem of a completely different kind. The
flavor of our applications is also completely different.
|
math
|
741 |
Interpolating sequences for weighted Bergman spaces of the ball
|
math.CV
|
Let $B_{\alpha}^{p}$ be the space of $f$ holomorphic in the unit ball of
$\Bbb C^n$ such that $(1-|z|^2)^\alpha f(z) \in L^p$, where $0<p\leq\infty$,
$\alpha\geq -1/p$ (weighted Bergman space). In this paper we study the
interpolating sequences for various $B_{\alpha}^{p}$. The limiting cases
$\alpha=-1/p$ and $p=\infty$ are respectively the Hardy spaces $H^p$ and
$A^{-\alpha}$, the holomorphic functions with polynomial growth of order
$\alpha$, which have generated particular interest.
In \S 1 we first collect some definitions and well-known facts about weighted
Bergman spaces and then introduce the natural interpolation problem, along with
some basic properties. In \S 2 we describe in terms of $\alpha$ and $p$ the
inclusions between $B_{\alpha}^{p}$ spaces, and in \S 3 we show that most of
these inclusions also hold for the corresponding spaces of interpolating
sequences. \S 4 is devoted to sufficient conditions for a sequence to be
$B_{\alpha}^{p}$-interpolating, expressed in the same terms as the conditions
given in previous works of Thomas for the Hardy spaces and Massaneda for
$A^{-\alpha}$. In particular we show, under some restrictions on $\alpha$ and
$p$, that finite unions of $B_{\alpha}^{p}$-interpolating sequences coincide
with finite unions of separated sequences.
In his article in Inventiones, Seip implicitly gives a characterization of
interpolating sequences for all weighted Bergman spaces in the disk. We spell
out the details for the reader's convenience in an appendix (\S 5).
|
math
|
742 |
Global $C^\nf$ Irregularity of the $\bar\partial$--Neumann Problem for Worm Domains
|
math.CV
|
No abstract available.
|
math
|
743 |
On Analytic Solvability and Hypoellipticity For $\dbar$ and $\dbar_b
|
math.CV
|
No abstract available.
|
math
|
744 |
Nonexistence of Continuous Peaking Functions
|
math.CV
|
We construct a smoothly bounded pseudoconvex domain such that every boundary
point has a p.s.h. peak function but some boundary point admits no (local)
holomorphic peak function.
|
math
|
745 |
The monodromy groups of Schwarzian equations on closed Riemann surfaces
|
math.CV
|
Let \theta:\pi_1(R) \to \PSL(2,\C) be a homomorphism of the fundamental group
of an oriented, closed surface R of genus exceeding one. We will establish the
following theorem.
Theorem. Necessary and sufficient for \theta to be the monodromy
representation associated with a complex projective stucture on R, either
unbranched or with a single branch point of order 2, is that \theta(\pi_1(R))
be nonelementary. A branch point is required if and only if the representation
\theta does not lift to \SL(2,\C).
|
math
|
746 |
Regularity of Holomorphic Correspondences and Applications to the Mapping Problems
|
math.CV
|
We study the regularity results of holomorphic correspondences. As an
application, we combine it with certain recently developed methods to obtain
the extension theorem for proper holomorphic mappings between domains with real
analytic boundaries in the complex 2-space.
|
math
|
747 |
An Example of a Domain with Non-Compact Automorphism Group
|
math.CV
|
We give an example of a bounded, pseudoconvex, circular domain in ${\mathbb
C}^3$ with smooth, real-analytic boundary and non-compact automorphism group,
which is not biholomorphically equivalent to any Reinhardt domain.
|
math
|
748 |
Examples of domains with non-compact automorphism groups
|
math.CV
|
We give, in dimensions three or greater, an example of a bounded,
pseudoconvex, circular domain in complex space with smooth real analytic
boundary and non-compact automorphism group which is not biholomorphically
equivalent to any Reinhardt domain. We give an analogous example in dimension
two, but the domain fails to be smooth at one boundary point---indeed it is not
in any Lipschitz class at the exceptional point.
|
math
|
749 |
On the product property of the pluricomplex Green function
|
math.CV
|
We prove that the pluricomplex Green function has the product property
$g_{D_1\times D_2}=\max\{ g_{D_1},g_{D_2}\}$ for any domains $D_1\subset\Bbb
C^n$ and $D_2\subset\Bbb C^m$.
|
math
|
750 |
Divergence of projective structures and lengths of measured laminations
|
math.CV
|
Given a complex structure, we investigate diverging sequences of projective
structures on the fixed complex structure in terms of Thurston's
parametrization. In particular, we will give a geometric proof to the theorem
by Kapovich stating that as the projective structures on a fixed complex
structure diverge so do their monodromies. In course of arguments, we extend
the concept of realization of laminations for PSL$(2,{\mathbf
C})$-representations of surface groups.
|
math
|
751 |
CR automorphisms of real analytic manifolds In complex space
|
math.CV
|
In this paper we shall give sufficient conditions for local CR
diffeomorphisms between two real analytic submanifolds of $\Bbb C^N$ to be
determined by finitely many derivatives at finitely many points. These
conditions will also be shown to be necessary in model cases. We shall also
show that under the same conditions, the Lie algebra of the infinitesimal CR
automorphisms at a point is finite dimensional.
|
math
|
752 |
Errata for Geometric Function Theory in Several Complex Variables
|
math.CV
|
This is a list of corrections for the book: J. Noguchi and T. Ochiai,
Geometric Function Theory in Several Complex Variables, xi + 282 pp., Math.\
Monographs Vol.\ {\bf 80}, Amer.\ Math.\ Soc., Providence, 1990. The authors
hope that this distribution will be helpful for readers to avoid unnecessary
confusions.
|
math
|
753 |
Nonvanishing of the differential of holomorphic mappings at boundary points
|
math.CV
|
In this paper we prove a general result of the ``Hopf lemma'' type for CR
mappings, with nonidentically vanishing Jacobians, between real hypersurfaces
in C^n with smooth or real analytic boundaries. Applications of this result to
finiteness and holomorphic extendibility of such mappings are also given. The
novelty here is that we make no assumption on the nonflatness of the mapping or
its Jacobian, nor do we assume that the hypersurfaces are pseudoconvex or
minimally convex.
|
math
|
754 |
An interpolation theorem for holomorphic automorphisms of {\bf C}$^n$
|
math.CV
|
We construct automorphisms of $\C^n$ which map certain discrete sequences one
onto another with prescribed finite jet at each point, thus solving a general
Mittag-Leffler interpolation problem for automorphisms. Under certain
circumstances, this can be done while also approximating a given automorphism
on a compact set.
|
math
|
755 |
On the defect of an analytic disc
|
math.CV
|
Although the concept of defect of an analytic disc attached to a generic
manifold of $\C^{n}$ seems to play a merely technical role, it turns out to be
a rather deep and fruitful notion for the extendability of CR functions defined
on the manifold.
In this paper we give a new geometric description of defect, drawing
attention to the behaviour of the interior points of the disc by infinitesimal
perturbations. For hypersurfaces a stronger result holds because these
perturbations describe a complex vector space of $\C^{n}$.
For a big analytic disc the defect does not need to be smaller than the
codimension of the manifold. Indeed we show by an example that it can be
arbitrarily large independently of the codimension of the manifold.
Nevertheless we also prove that the defect is always finite. In the case of a
hypersurface we give a geometric upper bound for the defect.
|
math
|
756 |
Defect and evaluations
|
math.CV
|
Let $S$ be a generic submanifold of $C^N$ of real codimension m. In this work
we continue the study, carried over by various authors, of the set of analytic
discs attached to S. Let $M$ be the set of analytic discs attached to $S.$
Given $q \in S$ let $M_q$ be the set of discs $\phi$ in M such that $\phi_(1).$
B. Trepreau and other authors gave sufficient conditions for $M$ to be a
manifold in a neighborhood of a given disc. We give conditions for $M_q$ to be
a manifold. When this conditions are satisfied we look at the map on $M$ given
by $\phi \rightarrow \phi(0),$ and we describe the image of its differential,
(in particular we determine its dimension). We then do the same for the map
$\phi \rightarrow \phi(-1)$ on $M_q.$ For example we find as a corollary that
if S has only minimal points, then there exists an open dense subset $Omega$ in
M such that the restriction of the map $\phi \rightarrow \phi(0)$ to $\Omega$
is an open map with value in $C^N.$
|
math
|
757 |
On totally real spheres in complex space
|
math.CV
|
We shall prove that there are totally real and real analytic embeddings of
$S^k$ in $\cc^n$ which are not biholomorphically equivalent if $k\geq 5$ and
$n=k+2[\frac{k-1}{4}]$. We also show that a smooth manifold $M$ admits a
totally real immersion in $\cc^n$ with a trivial complex normal bundle if and
only if the complexified tangent bundle of $M$ is trivial. The latter is proved
by applying Gromov's weak homotopy equivalence principle for totally real
immersions to Hirsch's transversal fields theory.
|
math
|
758 |
On the boundary orbit accumulation set for a domain with non-compact automorphism group
|
math.CV
|
For a smoothly bounded pseudoconvex domain $D\subset{\Bbb C}^n$ of finite
type with non-compact holomorphic automorphism group $\text{Aut}(D)$, we show
that the set $S(D)$ of all boundary accumulation points for $\text{Aut}(D)$ is
a compact subset of $\partial D$ and, if $S(D)$ contains at least three points,
it is connected and thus has the power of the continuum. We also discuss how
$S(D)$ relates to other invariant subsets of $\partial D$ and show in
particular that $S(D)$ is always a subset of the \v{S}ilov boundary.
|
math
|
759 |
A Carleman type theorem for proper holomorphic embeddings
|
math.CV
|
In 1927, Carleman showed that a continuous, complex-valued function on the
real line can be approximated in the Whitney topology by an entire function
restricted to the real line. In this paper, we prove a similar result for
proper holomorphic embeddings. Namely, we show that a proper $\cC^r$ embedding
of the real line into $\C^n$ can be approximated in the strong $\cC^r$ topology
by a proper holomorphic embedding of $\C$ into $\C^n$.
|
math
|
760 |
Fuchsian Groups, Quasiconformal Groups, and Conical Limit Sets
|
math.CV
|
We construct examples showing that the normalized Lebesgue measure of the
conical limit set of a uniformly quasiconformal group acting discontinuously on
the disc may take any value between zero and one. This is in contrast to the
cases of Fuchsian groups acting on the disc, conformal groups acting
discontinuously on the ball in dimension three or higher, uniformly
quasiconformal groups acting discontinuously on the ball in dimension three or
higher, and discrete groups of biholomorphic mappings acting on the ball in
several complex dimensions. In these cases the normalized Lebesgue measure is
either zero or one.
|
math
|
761 |
Generalized Bergmann Metrics and Invariance of Plurigenera
|
math.CV
|
An invariant kernel for the pluricanonical system of a projective manifold of
general type is introduced. Using this kernel we prove that the Yau volume form
on a smooth projective variety has seminegative Ricci curvature. As a biproduct
we prove the invariance of plurigenera for smooth projective deformations of
manifolds of general type.
|
math
|
762 |
Finitely smooth Reinhardt domains with non-compact automorphism group
|
math.CV
|
We give a complete description of bounded Reinhardt domains of finite
boundary smoothness that have non-compact automorphism group. As part of this
program, we show that the classification of domains with non-compact
automorphism group and having only finite boundary smoothness is considerably
more complicated than the classification of such domains that have infinitely
smooth boundary.
|
math
|
763 |
Singularities of the Bergman kernel for certain weakly pseudoconvex domains
|
math.CV
|
Consider the Bergman kernel $K^B(z)$ of the domain $\ellip = \{z \in \Comp^n
; \sum_{j=1}^n |z_j|^{2m_j}<1 \}$, where $m=(m_1,\ldots,m_n) \in \Natl^n$ and
$m_n \neq 1$. Let $z^0 \in \partial \ellip$ be any weakly pseudoconvex point,
$k \in \Natl$ the degenerate rank of the Levi form at $z^0$. An explicit
formula for $K^B(z)$ modulo analytic functions is given in terms of the polar
coordinates $(t_1, \ldots, t_k, r)$ around $z^0$. This formula provides
detailed information about the singularities of $K^B(z)$, which improves the
result of A. Bonami and N. Lohou\'e \cite{bol}. A similar result is established
also for the Szeg\"o kernel $K^S(z)$ of $\ellip$.
|
math
|
764 |
Bohr's power series theorem in several variables
|
math.CV
|
Generalizing a classical one-variable theorem of Harald Bohr, we show that if
an n-variable power series has modulus less than 1 in the unit polydisc, then
the sum of the moduli of the terms is less than 1 in the polydisc of radius
1/(3*n^{1/2}).
|
math
|
765 |
Breakdown of analyticity for d-bar-b and Szego kernels
|
math.CV
|
The CR manifold M_m = { Im z_2= Re z_1^{2m} } (m=2,3,...) is the
counterexample, which has been given by M. Christ and D. Geller, to analytic
hypoellipticity of d-bar-b and real analyticity of the Szego kernel. In order
to give a direct interpretation for the breakdown of real analyticity of the
Szego kernel, we give a Borel summation type representation of the Szego kernel
in terms of simple singular solutions of the equation d-bar-b u = 0.
|
math
|
766 |
The dbar-Neumann problem in the Sobolev topology
|
math.CV
|
revision posted November 1996
|
math
|
767 |
Gleason's problem in weighted Bergman space on egg domains
|
math.CV
|
In the paper, we discuss on the egg domains: $$
\Omega_a=\left\{\xi=(z,w)\in\bold C^{n+m}: \ z\in\bold C^n, \ w\in\bold C^m,
|z|^2+|w|^{2/a}<1\right\}, \qquad 0<a\le 2. $$ We show that Gleason's problem
can be solved in the weight Bergman space on theegg domains. The proof will
need the help of the recent work of the second named author on the weighted
Bergman projections on this kind of domain. As an application, we obtain a
multiplier theorem on the egg domains.
|
math
|
768 |
Effective formulas for invariant functions -- case of elementary Reinhardt domains
|
math.CV
|
In the paper we find effective formulas for the invariant functions,
appearing in the theory of several complex variables, of the elementary
Reinhardt domains. This gives us the first example of a large family of domains
for which the functions are calculated explicitly.
|
math
|
769 |
Sampling sets for Hardy spaces of the disk
|
math.CV
|
We propose two possible definitions for the notion of a sampling sequence (or
set) for Hardy spaces of the disk. The first one is inspired by recent work of
Bruna, Nicolau, and \O yma about interpolating sequences in the same spaces,
and it yields sampling sets which do not depend on the value of $p$ and
correspond to the result proved for bounded functions ($p=\infty$) by Brown,
Shields and Zeller. The second notion, while formally closer to the one used
for weighted Bergman spaces, is shown to lead to trivial situations only, but
raises a possibly interesting problem.
|
math
|
770 |
Asymptotic expansion of the Bergman kernel for weakly pseudoconvex tube domains in C^2
|
math.CV
|
In this paper we give an asymptotic expansion of the Bergman kernel for
certain weakly pseudoconvex tube domains of finite type in C^2. Our asymptotic
formula asserts that the singularity of the Bergman kernel at weakly
pseudoconvex points is essentially expressed by using two variables; moreover
certain real blowing-up is necessary to understand its singularity. The form of
the asymptotic expansion with respect to each variable is similar to that in
the strictly pseudoconvex case due to C. Fefferman. We also give an analogous
result in the case of the Szego kernel.
|
math
|
771 |
Sharp Lipschitz estimates for operator dbar_M on a q-concave CR manifold
|
math.CV
|
We prove that the integral operators $R_r$ and $H_r$ constructed in \cite{P}
and such that $$f = \bar\partial_{\bold M} R_r(f) + R_{r+1}(\bar\partial_{\bold
M} f) + H_r(f),$$ for a differential form $f \in C_{(0,r)}^{\infty}({\bold M})$
on a regular q-concave CR manifold ${\bold M}$ admit sharp estimates in the
Lipschitz scale.
|
math
|
772 |
Defects for Ample Divisors of Abelian Varieties, Schwarz Lemma, and Hyperbolic Hypersurfaces of Low Degrees
|
math.CV
|
The main purpose of this paper is to prove the following theorem on the
defect relations for ample divisors of abelian varieties.
Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$
be an ample divisor in $A$. Let $f:{\bf C}\rightarrow A$ be a holomorphic map.
Then the defect for the map $f$ and the divisor $D$ is zero.
Corollary to Main Theorem. The complement of an ample divisor $D$ in an
abelian variety $A$ is hyperbolic in the sense that there is no nonconstant
holomorphic map from $\bf C$ to $A-D$.
|
math
|
773 |
Hyperbolic Reinhardt Domains in C^2 with Noncompact Automorphism Group
|
math.CV
|
We give an explicit description of hyperbolic Reinhardt domains D in C^2 such
that: (i) D has C^k-smooth boundary for some k greater than or equal to 1, (ii)
D intersects at least one of the coordinate complex lines $\{z_1=0\}$,
$\{z_2=0\}$, and (iii) D has noncompact automorphism group. We also give an
example that explains why such a setting is natural for the case of hyperbolic
domains and an example that indicates that the situation in C^n for n greater
than or equal to 3 is essentially more complicated than that in C^2.
|
math
|
774 |
The involutive structure on the blow-up of R^n in C^n
|
math.CV
|
We investigate the natural involutive structure on the blow-up of ${\Bbb
R}^n$ in ${\Bbb C}^n$ extending the complex structure on the complement of the
exceptional hypersurface. Our main result is that this structure is
hypocomplex, meaning that any solution is locally a holomorphic function of a
basic set of independent solutions. We show this by an elementary power series
argument but note that the result is essentially equivalent to the Edge of the
Wedge Theorem. In particular, we obtain a relatively simple new proof of this
classical theorem.
|
math
|
775 |
Application of the Complex Monge-Ampere equation to the study of proper holomorphic mappings of strictly pseudoconvex domains
|
math.CV
|
We construct a special plurisubharmonic defining function for a smoothly
bounded strictly pseudoconvex domain so that the determinant of the complex
Hessian vanishes to high order on the boundary. This construction, coupled with
regularity of solutions of complex Monge-Ampere equation and the reflection
principle, enables us to give a new proof of the Fefferman mapping theorem.
|
math
|
776 |
$\overline{\partial}$-Neumann Problem in the Sobolev Topology
|
math.CV
|
We study the $\overline{\partial}$-Neumann problem using the Sobolev space
inner product. We show that the problem can be solved on any smoothly bounded,
pseudoconvex domain. We further formulate estimates and the basic results of a
Sobolev Hodge theory.
|
math
|
777 |
Domains with Non-Compact Automorphism Group: A Survey
|
math.CV
|
We survey results arising from the study of domains in C^n with non-compact
automorphism group. Beginning with a well-known characterization of the unit
ball, we develop ideas toward a consideration of weakly pseudoconvex (and even
non-pseudoconvex) domains with particular emphasis on characterizations of (i)
smoothly bounded domains with non-compact automorphism group and (ii) the Levi
geometry of boundary orbit accumulation points.
Particular attention will be paid to results derived in the past ten years.
|
math
|
778 |
The Julia-Wolff-Caratheodory theorem in polydisks
|
math.CV
|
The classical Julia-Wolff-Caratheodory 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 in C^n, and by the author to holomorphic maps between strongly
(pseudo)convex domains. Here we describe Julia-Wolff-Caratheodory 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 main tool for
the proof is a general version of Lindelof's principle valid for not
necessarily bounded holomorphic functions.
|
math
|
779 |
Global regularity of the dbar-Neumann problem: a survey of the L^2-Sobolev theory
|
math.CV
|
This is a survey article written for the proceedings of the special year in
several complex variables, 1995-1996, at the Mathematical Sciences Research
Institute in Berkeley.
|
math
|
780 |
Plurisubharmonic functions and subellipticity of the dbar-Neumann problem on nonsmooth domains
|
math.CV
|
We show subellipticity of the d-bar Neumann problem on domains with Lipschitz
boundary in the presence of plurisubharmonic functions with Hessians of
algebraic growth. In particular, a subelliptic estimate holds near a point
where the boundary is piecewise smooth of finite type.
|
math
|
781 |
Parametrization of local biholomorphisms of real analytic hypersurfaces
|
math.CV
|
Let $M$ be a real analytic hypersurface in $\bC^N$ which is finitely
nondegenerate, a notion that can be viewed as a generalization of Levi
nondegenerate, at $p_0\in M$. We show that if $M'$ is another such hypersurface
and $p'_0\in M'$, then the set of germs at $p_0$ of biholomorphisms $H$ with
$H(M)\subset M'$ and $H(p_0)=p'_0$, equipped with its natural topology, can be
naturally embedded as a real analytic submanifold in the complex jet group of
$\bC^N$ of the appropriate order. We also show that this submanifold is defined
by equations that can be explicitly computed from defining equations of $M$ and
$M'$. Thus, $(M,p_0)$ and $(M',p'_0)$ are biholomorphically equivalent if and
only if this (infinite) set of equations in the complex jet group has a
solution.
Another result obtained in this paper is that any invertible formal map $H$
that transforms $(M,p_0)$ to $(M',p'_0)$ is convergent. As a consequence,
$(M,p_0)$ and $(M',p'_0)$ are biholomorphically equivalent if and only if they
are formally equivalent.
|
math
|
782 |
Blow-analytic retraction onto the central fibre
|
math.CV
|
Let X be a complex analytic space and let f:X -> C be a proper complex
analytic function with nonsingular generic fibres. By adapting the blowanalytic
methods of Kuo we construct a retraction of a neighbourhood of the central
fibre f^{-1}(0) onto f^{-1}(0). Our retraction is defined by the flow of a real
analytic vector field on an oriented real analytic blow-up of X. Then we
describe in terms of this blow-up the associated specialization map and local
Milnor fibrations. The method also works in real analytic category.
|
math
|
783 |
Positivity conditions for bihomogeneous polynomials
|
math.CV
|
In this paper we continue our study of a complex variables version of
Hilbert's seventeenth problem by generalizing some of the results from [CD].
Given a bihomogeneous polynomial $f$ of several complex variables that is
positive away from the origin, we proved that there is an integer $d$ so that
$||z||^{2d} f(z,{\overline z})$ is the squared norm of a holomorphic mapping.
Thus, although $f$ may not itself be a squared norm, it must be the quotient of
squared norms of holomorphic homogeneous polynomial mappings. The proof
required some operator theory on the unit ball. In the present paper we prove
that we can replace the squared Euclidean norm by squared norms arising from an
orthonormal basis for the space of homogeneous polynomials on any bounded
circled pseudoconvex domain of finite type. To do so we prove a compactness
result for an integral operator on such domains related to the Bergman kernel
function.
|
math
|
784 |
Multiplicity of a zero of an analytic function on a trajectory of a vector field
|
math.CV
|
Let P(x) be a germ at the origin of an analytic function in C^n, where x =
(x_1,..., x_n), and let
\xi = \xi_1(x) d/dx_1 + ... + \xi_n(x) d/dx_n
be a germ at the origin of an analytic vector field. Suppose that \xi(0) !=
0, and let \gamma be a trajectory of \xi through the origin. Suppose that
P|_\gamma /\equiv 0, and let \mu(P|_\gamma) be the multiplicity of a zero of
P|_\gamma at the origin. Let
\xi P = \xi_1 dP/dx_1 + ... + \xi_n dP/dx_n
be derivative of P in the direction of \xi, and let \xi^kP be the kth
iteration of this derivative.
We give a formula (Theorem 1) for \mu(P|_\gamma) in terms of the Euler
characteristic of the Milnor fibers defined by a deformation of P, \xi P, ...,
\xi^{n-1}P . For a polynomial P of degree p and a vector field \xi with
polynomial coefficients of degree q, this allows one to compute \mu(P|_\gamma)
in purely algebraic terms (Theorem 2), and to give an estimate (Theorem 3) for
\mu(P|_\gamma) in terms of n, p, q, single exponential in n and polynomial in p
and q. This estimate improves previous results which were doubly exponential in
n.
|
math
|
785 |
On the Lojasiewicz exponent of the gradient of a polynomial function
|
math.CV
|
Let h = \sum h_{\alpha \beta} X^\alpha Y^\beta be a polynomial with complex
coefficients. The Lojasiewicz exponent of the gradient of h at infinity is the
upper bound of the set of all real \lambda such that |grad h(x, y)| >=
c|(x,y)|^\lambda in a neighbourhood of infinity in C^2, for c > 0. We estimate
this quantity in terms of the Newton diagram of h. The equality is obtained in
the nondegenerate case.
|
math
|
786 |
Factorization of proper holomorphic mappings through Thullen Domains
|
math.CV
|
In this article, we consider a bounded pseudoconvex domain in ${\bf C}^2$
satifying:
(a) it admits a proper holomorphic mapping $f$ onto the unit ball $B^2$, and
(b) it is simply connected and has a real analytic boundary.
According to [Barletta-Bedford, Indiana U. Math. J, 39(1985), 315-338], the
strong pseudconvexity of $B^2$ alone yields that such a domain is "weakly
spherical" at the boundary points that are at the same time a smooth point of
the branch locus $Z_{df} = \{\det(J_{\bf C} f) = 0\}$. (Notice that
[Diederich-Fornaess, Math. Ann., 282 (1988), 681-700] implies that $f$ as well
as $Z_{df}$ extends holomorphically across the boundaries.)
Our main contribution in this paper is that we have discovered a stronger
rigidity (both local and global) in case the target domain is the unit ball.
The main results are:
THEOREM ("Local Rigidity"): Let $(M,o)$ be a real analytic normalized weakly
spherical pointed CR hypersurface in ${\bf C}^2$ of order $k_0 > 1$. Let
$(\Sigma, o)$ be the pointed Siegel hypersurface given by the defining equation
$Re w - |z|^2 = 0$. If there is a holomorphic mapping $F:(M,o) \to (\Sigma,o)$
for which $o$ is a regular branch point, then
(1) $(M,o)$ is defined by the equation $Re w - |z|^{2k_0} = 0$, and
(2) $F(z,w)$ is equivalent to $(z,w) \mapsto (z^{k_0},w)$ up to a composition
with elements in $Aut (M,o)$ and $Aut (\Sigma,o)$.
THEOREM ("Global Rigidity"): Let $D$ and $f:D \to B^2$ be as above, and let
$f$ be generically $m$-to-1. Assume that its branch locus $Z_{df}$ admits an
analytic component $V$ with the following properties:
(1) $f$ is locally a $m$-to-1 branched covering with branch locus $V$ at
every point of $V \cap \partial D$;
(2) $V \cap \partial D$ is connected and contains no singular point of the
variety $Z_{df}$.
Then $D$ is biholomorphic to $E_m = \{|z|^{2m} + |w|^2 < 1\}$.
|
math
|
787 |
Extension Properties of Meromorphic Mappings with Values in Non-Kahler Manifolds
|
math.CV
|
We prove an analogue of E. Levi's Continuity Principle for meromorphic
mappings with values in arbitrary compact complex manifolds in place of the
Riemann sphere $\cc\pp^1$. The result is achieved by introducing a new
extension method for meromorphic mappings. One of the corollaries reads as
follows:
If a compact complex surface $X$ is not "among the known ones" then for every
domain $\Omega $ in a Stein surface every meromorphic mapping $f:\Omega \to X$
is in fact holomorphic and extends as a holomorphic mapping $\hat f:\hat D\to
X$ of the envelope of holomorphy $\hat D$ of $D$ into $X$.
In this last version also two examples of compact complex maniflds are
described with meromoprhic mappings into these manifolds having thin but
non-analytic singularity sets.
|
math
|
788 |
Normal forms and biholomorphic equivalence of real hypersurfaces in C^3
|
math.CV
|
We consider the problem of describing the local biholomorphic equivalence
class of a real-analytic hypersurface $M$ at a distinguished point $p_0\in M$
by giving a normal form for such objects. In order for the normal form to carry
useful information about the biholomorphic equivalence class, we shall require
that the transformation to normal form is unique modulo some finite dimensional
group. A classical result due to Chern--Moser gives such a normal form for Levi
nondegenerate hypersurfaces.
The main results in this paper concern real-analytic hypersurfaces $M$ in
$\Bbb C^3$ at certain Levi degenerate points $p_0\in M$, namely points at which
$M$ is 2-nondegenerate. We give a partial normal form for all such $(M,p_0)$,
i.e. a normal form for the data associated with 2-nondegeneracy. We also give a
complete formal normal form for such $(M,p_0)$ under the additional condition
that the Levi form has rank one at $p_0$. This result, combined with a recent
theorem due to the author, M. S. Baouendi, and L. P. Rothschild stating that
formal equivalences between real-analytic finitely nondegenerate hypersurfaces
converge, gives a description of the biholomorphic equivalence class of a
real-analytic hypersurface in $\Bbb C^3$ at a point of 2-nondegeneracy where
the rank of the Levi form is one.
|
math
|
789 |
Optimal regularity for d-bar-b on CR manifolds
|
math.CV
|
In this paper a new explicit integral formula is derived for solutions of the
tangential Cauchy-Riemann equations on CR q-concave manifolds and optimal
estimates in the Lipschitz norms are obtained.
|
math
|
790 |
Some applications of the Kohn-Rossi extension theorems
|
math.CV
|
We prove extension results for meromorphic functions by combining the
Kohn-Rossi extension theorems with Andreotti's theory on the algebraic and
analytic dependence of meromorphic functions on pseudoconcave manifolds.
Versions of Kohn-Rossi theorems for pseudoconvex domains are included.
|
math
|
791 |
The football player and the infinite series
|
math.CV
|
This is the text of an expository talk given at the May 1997 Detroit meeting
of the American Mathematical Society. It is a tale of a famous football player
and a subtle problem he posed about the uniform convergence of Dirichlet
series. Hiding in the background is the theory of analytic functions of an
infinite number of variables.
|
math
|
792 |
Zeta-functions for germs of meromorphic functions and Newton diagrams
|
math.CV
|
For a germ of a meromorphic function f=P/Q, we offer notions of the monodromy
operators at zero and at infinity. If the holomorphic functions P and Q are
non-degenerated with respect to their Newton diagrams, we give an analogue of
the formula of Varchenko for the zeta-functions of these monodromy operators.
|
math
|
793 |
Ordinary differential equations with only entire solutions
|
math.CV
|
We prove necessary and sufficient conditions for a system $\dot
z_i=z_ip_i(z)$ ($p_i$ a polynomial) to have only entire analytic functions as
solutions.
|
math
|
794 |
The Bergman kernel function: explicit formulas and zeroes
|
math.CV
|
We show how to compute the Bergman kernel functions of some special domains
in a simple way. As an application of the explicit formulas, we show that the
Bergman kernel functions of some convex domains, for instance the domain in C^3
defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes.
|
math
|
795 |
On the Lojasiewicz exponent at infinity for polynomial functions
|
math.CV
|
The Lojasiewicz exponent at infinity of an entire function measures of the
infimal rate of growth of its gradient. The authors compute the Lojasiewicz
exponents at infinity of the 3-variable complex polynomials
x - 3 x^{2n+1} y^{2q} + 2 x^{3n+1} y^{3q} + y z
|
math
|
796 |
Holomorphic factorization of matrices of polynomials
|
math.CV
|
This paper considers some work done by the author and Catlin [CD1,CD2,CD3]
concerning positivity conditions for bihomogeneous polynomials and metrics on
bundles over certain complex manifolds. It presents a simpler proof of a
special case of the main result in [CD3], providing also a self-contained proof
of a generalization of the main result from [CD1]. Some new examples and
applications appear here as well. The idea is to use the Bergman kernel
function and some operator theory to prove purely algebraic theorems about
matrices of polynomials.
Theorem 1. [Catlin-D'Angelo]. Suppose that $f$ is a bihomogeneous real-valued
polynomial on ${\bf C^n}$ of degree $2m$. Then $f$ is positive away from the
origin if and only there is an integer $d$ and a holomorphic homogeneous
polynomial mapping $A$, whose components span the space of homogeneous
polynomials of degree $m+d$, such that $$ ||z||^{2d} f(z,{\overline z}) =
||A(z)||^2.$$ Suppose that $F(z,{\overline z})$ is an $r$ by $r$ matrix whose
entries are bihomogeneous polynomials of degree $2m$. Then $F(z,{\overline z})$
is positive-definite at each point $z \ne 0$ if and only if there is an integer
$d$ and a holomorphic homogeneous polynomial matrix $A$, whose row vectors span
the space of $r$-tuples of homogeneous polynomials of degree $m+d$, such that
$$||z||^{2d} F(z,{\overline z}) = A(z)^* A(z).$$
|
math
|
797 |
On a domain in C^2 with generic piecewise smooth Levi-flat boundary and non-compact automorphism group
|
math.CV
|
In this paper, we prove that if D is a simply-connected domain in C^2 with
generic piecewise smooth Levi-flat boundary and non-compact automorphism group,
then D is biholomorphic to the bidisc. The proof is based on a careful analysis
of invariant measures.
|
math
|
798 |
Compactness of the d-bar-Neumann problem on convex domains
|
math.CV
|
The d-bar-Neumann operator on (0,q)-forms ($1\le q \le n$) on a bounded
convex domain Omega in C^n is compact if and only if the boundary of Omega
contains no complex analytic (equivalently: affine) variety of dimension
greater than or equal to q.
|
math
|
799 |
Finite interpolation with minimum uniform norm in C^n
|
math.CV
|
Given a finite sequence $a:={a_1, ..., a_N}$ in a domain $\Omega \subset
C^n$, and complex scalars $v:={v_1, ..., v_N}$, consider the classical extremal
problem of finding the smallest uniform norm of a holomorphic function
verifying $f(a_j)=v_j$ for all $j$. We show that the modulus of the solutions
to this problem must approach its least upper bound along a subset of the
boundary of the domain large enough to contain the support of a measure whose
hull contains a subset of the original $a$ large enough to force the same
minimum norm. Furthermore, all the solutions must agree on a variety which also
contains this hull. An example is given to show that the inclusions can be
strict.
|
math
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.