Feuilletages holomorphes locaux et analyticité partielle d'applications C ∞

 Let f : M → M′ be a smooth CR mapping between a generic real analytic submanifold M ⊂ ℂ ⁿ , n > 1, and a real analytic subset M′ ⊂ ℂ ^n′ . We prove that if M is minimal and if M′ does not contain any complex curves, then f is analytic on a dense open subset of M. More generally, we establish an upper estimate of the partial analyticity of f, which depends on the maximal dimension of local holomorphic foliations contained in M ^′ . Received: 7 August 2001 Mathematics Subject Classification (2000): 32V25, 32V40, 32H99     

Analytic continuation of holomorphic correspondences and equivalence of domains in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

The following result is proved: Let D and D′ be bounded domains in ℂ ⁿ , ∂D is smooth, real-analytic, simply connected, and ∂D′ is connected, smooth, real-algebraic. Then there exists a proper holomorphic correspondence f:D→D′ if and only if there exist points p∈∂D and p′∈∂D′, such that ∂D and ∂D′ are locally CR-equivalent near p and p′. This leads to a characterization of the equivalence relationship between bounded domains in ℂ ⁿ modulo proper holomorphic correspondences in terms of local CR-equivalence of their boundaries. Oblatum 23-I-2002 & 18-XI-2002?Published online: 17 February 2003     

Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties

Let X ⊂ ℂⁿ be a smooth affine variety of dimension n – r and let f = (f₁,..., f_m): X → ℂ_m be a polynomial dominant mapping. We prove that the set K(f) of generalized critical values of f (which always contains the bifurcation set B(f) of f) is a proper algebraic subset of ℂ_m. We give an explicit upper bound for the degree of a hypersurface containing K(f). If I(X)—the ideal of X—is generated by polynomials of degree at most D and deg f_i ≤ d, then K(f) is contained in an algebraic hypersurface of degree at most (d + (m – 1)(d – 1)+(D – 1)r)_n-rD_r. In particular if X is a hypersurface of degree D and f: X → ℂ is a polynomial of degree d, then f has at most (d + D – 1)_n-1D generalized critical values. This bound is asymptotically optimal for f linear. We give an algorithm to compute the set K(f) effectively. Moreover, we obtain similar results in the real case.     

A local extension theorem for proper holomorphic mappings in ℂ<Superscript>2</Superscript>

Let ƒ: D → D′ be a proper holomorphic mapping between bounded domains D, D′ in ℂ².Let M, M′ be open pieces on δD, δD′, respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under ƒ is contained in M′. It is shown that ƒ extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in ℂ².     

Boundary regularity of correspondences in ℂn

LetM, M′ be smooth, real analytic hypersurfaces of finite type in ℂⁿ and a holomorphic correspondence (not necessarily proper) that is defined on one side ofM, extends continuously up toM and mapsM to M′. It is shown that must extend acrossM as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich-Pinchuk extension result for CR maps.     

CR Hypersurfaces with a Contracting Automorphism

Let M be a real analytic CR hypersurface in ℂ ⁿ⁺¹ admitting no varieties of positive dimension. We show first that every contracting local CR automorphism of M is linearizable. As a consequence, we show that such M admitting a contracting local CR automorphism is holomorphically equivalent to a weighted homogeneous hypersurface. Finally, we apply these results to prove that a bounded domain in ℂ ⁿ⁺¹ with a real analytic boundary admitting an automorphism contracting at a boundary point must admit a Lie subgroup of real dimension at least two in its automorphism group. Research of the first named author is partially supported by The Grant R01-2005-000-10771-0 of The Korea Science and Engineering Foundation.     

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

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

On regular polynomial endomorphisms of ℂ<Superscript>2</Superscript> without bounded critical orbitswithout bounded critical orbits

We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ²↦ℂ² under which all complete external rays from infinity for f have well defined endpoints.     

Holomorphic functions which are highly nonintegrable at the boundary

Given a bounded convex domainD in ℂ^N with smooth boundary and a positive continuous function ϕ ond, it is proved that there is a holomorphic functionf onD such that |f|ϕ is nonintegrable onM∩D wheneverM is a real submanifold of a neighbourhood of a point ofbD which intersectsbD transversely.     

Bergman kernel and complex singularity exponent

We give a precise estimate of the Bergman kernel for the model domain defined by Ω _F = “(z,w) ∈ ℂ ⁿ⁺¹: Im w − |F(z)|² > 0”, where F = (f ₁, ..., f _m ) is a holomorphic map from ℂ ⁿ to ℂ ^m , in terms of the complex singularity exponent of F.     

On the dynamics of the singularly perturbed rational maps

In this paper, we study the dynamics of the family of rational maps fλ,（z） = zn - λ/zm, n ≥2, m ≥ 1,λ ∈ C. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of fλ.     

Fuzzy complex Grassmannians and quantization of line bundles

We construct by purely representation-theoretic methods fuzzy versions of an arbitrary complex Grassmannian M=Gr _n (ℂ ^n+m ), i.e., a sequence of matrix algebras tending SU(n+m)-equivariantly to the algebra of smooth functions on M. We also show that this approximation can be interpreted in terms of the Berezin-Toeplitz quantization of M. Furthermore, we use branching rules to prove that the quantization of every complex line bundle over M is given by a SU(n+m)-equivariant truncation of the space of its L ²-sections.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Realisierungen des idealen Randes einer Riemannschen Fläche unter konformen Abschließungen

Let a noncompact Riemann surface R of positive finite genus g be given. If f : R → R′ is a conformal mapping of R into a compact Riemann surface R′ of genus g, we have a realization of the ideal boundary of R on the surface R′. We consider (for the fixed R) all the possible R′ and the associated conformal mappings, and study how large the realized boundary can be. To this aim we pass to the (common) universal space ℂ ^g of the Jacobi variety of any R′ and show that the image sets of the ideal boundary of R in ℂ ^g are uniformly bounded.

     

Finite jet determination of CR embeddings

We prove finite jet determination results for smooth CR embeddings, which are of constant degeneracy, using the method of complete systems. As an application, we obtain a reflection principle for mappings between a Levinondegenerate hypersurface in ℂ ^N and a Levinondegenerate hypersurface in ℂ ^N+1.We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric [14].     

Problemi variazionali conformi

Letf: (M,g)→(N,g′) be a differentiable map between the riemannian manifoldsM andN, M being compact.K. Uhlenbeck pointed out a functionalE ^m(f), related to the energy density off, that depends only on the conformal structure ofM. In this paper we prove thatE ^m(f) is stationary with respect to deformations of the riemannian metric ofM if and only iff is weakly conformal; in this casef provides a local minimum ofE ^m.     

Integrable Submanifolds in Almost Complex Manifolds

Let (M,J) be a germ of an almost complex manifold of real dimension 2m and let n (n<m) be an integer. We study a necessary and sufficient condition for M to admit an integrable submanifold N of complex dimension n. If n=m−1, we find defining functions of N explicitly from the coefficients of the torsion tensor. For J obtained by small perturbation of the standard complex structure of ℂ ^m this condition is given as an overdetermined system of second order PDEs on the perturbation. The proof is based on the rank conditions of the Nijenhuis tensor and application of the Newlander-Nirenberg theorem. We give examples of almost complex structures on ℂ³: the ones with a single complex submanifold of dimension 2 and the ones with 1-parameter or 2-parameter families of complex submanifolds of dimension 2.     

On the range of the derivative of Gâteaux-smooth functions on separable banach spaces

We prove that there exists a Lipschitz function froml ¹ into ℝ² which is Gateaux-differentiable at every point and such that for everyx, y εl ¹, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0. This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198.     

Embeddings through discrete sets of balls

We investigate whether a Stein manifold M which allows proper holomorphic embedding into ℂ ⁿ can be embedded in such a way that the image contains a given discrete set of points and in addition follow arbitrarily close to prescribed tangent directions in a neighbourhood of the discrete set. The infinitesimal version was proven by Forstnerič to be always possible. We give a general positive answer if the dimension of M is smaller than n/2 and construct counterexamples for all other dimensional relations. The obstruction we use in these examples is a certain hyperbolicity condition.     

Equivalence of quotient Hilbert modules

LetM be a Hilbert module of holomorphic functions over a natural function algebraA(Ω), where Ω ⊆ ℂ ^m is a bounded domain. LetM ₀ ⊆M be the submodule of functions vanishing to orderk on a hypersurfaceZ ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modulesQ =M ⊖M ₀ The invariants are given explicitly in the particular case ofk = 2.