Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables

Generalized Analytic Functions. By I. N. Vekua. English Translation Editor I. N. Sneddon. Pergamon Press. 1962. Pp. 668. 105s.

We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group of a (suitably nondegenerate) real-analytic generic submanifold is a real algebraic Lie group. Moreover, it is shown that a formal equivalence between two such real-analytic submanifolds is necessarily convergent.

Rational dependence of smooth and analytic CR mappings on their jets

Results on finite determination and convergence of formal mappings between smooth generic submanifolds inC N are established in this article. The finite determination result gives suffi- cient 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 inC 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. 1. Introduction and main results In this article, we study formal mappings between smooth generic submanifolds in C N and establish results on finite determination, convergence and local biholomorphic, and algebraic equivalence. Our finite determination result gives sufficient conditions to guarantee that a formal map as above is uniquely determined by its jet (of a preassigned order) at a point. For real-analytic generic submanifolds, we prove convergence of formal mappings under appropriate assumptions and also give natural geometric conditions to assure that if two germs of such submanifolds are formally equivalent, then they are necessarily biholomorphically equivalent. If the submanifolds are moreover real-algebraic, we address the question of deciding when biholomorphic equivalence implies algebraic equivalence. In particular, we prove that if two real-algebraic hypersurfaces in C N are biholomorphically equivalent, then they are in fact 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. We now give precise definitions in order to state some of our main results. Let p 2 C N and

/pdf/reflection-ideals-and-mappings-between-generic-submanifolds-1orw5qvmnh.pdf

Reflection Ideals and mappings between generic submanifolds in complex space

Nous etablissons une estimee inferieure pour la metrique de Kobayashi-Royden sur une variete presque complexe (M,J) admettant une fonction bornee strictement pluri-sous-harmonique. Nous appliquons ce resultat a l'etude du comportement de la metrique au bord d'un domaine strictement pseudoconvexe dans M et donnons une condition suffisante d'hyperbolicite complete d'un domaine dans (M,J).

/pdf/estimates-of-the-kobayashi-royden-metric-in-almost-complex-3jn20nu7p5.pdf

Estimates of the Kobayashi-Royden metric in almost complex manifolds

Holomorphic maps of algebraic CR manifolds

We study some special almost complex structures on strictly pseudoconvex domains. They appear naturally as limits under a nonisotroping scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.

On the geometry of model almost complex manifolds with boundary

We study some special almost complex structures on strictly pseudoconvex domains in ℝ2n. They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

Riemann maps in almost complex manifolds

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Papers

Estimates of the Kobayashi-Royden metric in almost complex manifolds

Holomorphic maps of algebraic CR manifolds

On the geometry of model almost complex manifolds with boundary

On the geometry of model almost complex manifolds with boundary

Riemann maps in almost complex manifolds