Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ ^N are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂ ^N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.     

On the degree growth of birational mappings in higher dimension

Let ƒ be a birational map of C ^d ,and consider the degree complexity or asymptotic degree growth rate δ(ƒ) = lim_n → ∞ (deg(ƒⁿ))^1/n.We introduce a family of elementary maps, which have the form ƒ = L o J, where L is (invertible) linear, and J(x ₁ ⁻¹ ,..., x_d) = (x ₁ ⁻¹ ,...,x _d ⁻¹ .We develop a method of regularization and show how it can be used to compute δ for an elementary map.     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

Algebraicity of real analytic hypersurfaces and blowing-down

We show that a germ (M, 0) of real analytic strongly pseudoconvex hypersurface is holomorphically equivalent to a germ of real algebraic hypersurface if and only if there is a blowing-down from (M, 0) to a germ of real algebraic hypersurface.     

A reflection principle for real—Analytic submanifolds of complex spaces

We give an invariant nondegeneracy condition for CR-maps between generic submanifolds in different dimensions and use it to prove a reftection principle for these maps.     

Number-theoretic properties of certain CR mappings

We first prove a uniqueness result for certain group-invariant CR mappings to hyperquadrics. For cyclic groups these mappings lead to a collection of polynomials ƒ_p,q (with integer coefficients) in two variables; here p and q are positive integers. We use the uniqueness result to prove some surprising number-theoretic results about the ƒ_p,q, in particular, ƒ_p,q is congruent to x^P + y^P modulo (p) (for P ≥ 2) if and only if p is prime. We also determine recurrence relations for these polynomials for q ≤ 3 and determine a functional equation for a generating function. Finally, we discuss the invariant polynomials that arise for certain representations of dihedral groups to illustrate the non-Abelian case.     

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.     

Non-Autonomous basins of attraction and their boundaries

We study whether the basin of attraction of a sequence of automorphisms of ℂ ^{k is biholomorphic to} ℂ ^{k. In particular, we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to} ℂ ^{k, if every map is iterated sufficiently many times. We also construct Fatou-Bieberbach domains in} ℂ² whose boundaries are four-dimensional.     

An interpolation theorem for holomorphic automorphisms ofC n

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.     

Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group

We present a characterization of the open unit ball in a separable infinite dimensional Hilbert space by the property of automorphism orbits among the domains that are not necessarily bounded. This generalizes the recent work of Kim and Krantz [6]. Key new features of this article include: a lower bound estimate of the Kobayashi metric and distance near a pluri-subharmonic peak boundary point of the domains in Banach spaces, an effective localization argument, and an improvement of weak-type convergence of sequences of biholomorphic mappings of domains in Banach spaces.     

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderón-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on ℓ²-valued extensions of bounded linear operators is also obtained.     

In polytopes,small balls about some vertex minimize perimeter

In (the surface of) a convex polytope Pⁿ in ℝⁿ⁺¹, for small prescribed volume, geodesic balls about some vertex minimize perimeter.     

Diffeomorphisms of Stein structures

We prove that a pseudoholomorphic diffeomorphism between two almost complex manifolds with boundaries satisfying some pseudoconvexity type conditions cannot map a pseudoholomorphic disc in the boundary to a single point. This can be viewed as an almost complex analogue of a well known theorem of J. E. Fornaess.     

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].     

Existence of generic homoclinic tangencies for henon mappings

It is an open question whether there is a strange attractor for the Henon mapping. We show that for certain maps close to the Henon map there are strange attractors.     

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.     

Localization principle of automorphisms on generalized pseudoellipsoids

We deduce global biholomorphic equivalence from local biholomorphic equivalence near boundary points in the class of generalized pseudoellipsoids, extending local biholomorphisms to global ones. We discuss related results on holomorphic proper maps.     

Blaschke products and proper holomorphic mappings

Suppose f: $\mathbb{D} \to V$ is a proper holomorphic map of the unit disk $\mathbb{D} \subset \mathbb{C}$ onto a subset $V \subset \mathbb{D}$ of degree d>0. We show that f is conjugate to either an affine map or a degree d Blaschke product. As an application we give a unified treatment of theorems of Böttcher and Schröder coordinates.     

Distributional estimates for the bilinear Hilbert transform

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known L^p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L^p1 × L^p2 when either 1/p₁ + 1/p₂ = 3/2 or one of p₁, p₂ is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.     

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.