Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Function theory in several complex variables

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.

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Entire Functions Of Several Complex Variables

We study the Dirichlet problem for the Monge–Ampere equation on almost complex manifolds. We obtain the existence of the unique smooth solution in strictly pseudoconvex domains.

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The Monge–Ampère equation on almost complex manifolds

We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in $$\mathbb{C }^{n+1}$$
 , which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class $$\mathcal{C }^4$$
 . This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in the almost complex setting.

Stationary holomorphic discs and finite jet determination problems

Stationary discs and finite jet determination for non-degenerate generic real submanifolds

Let ( ·,�) be smooth, i.e. C 1 , embeddings from onto � , where and � are bounded domains with smooth boundary in the complex plane andvaries in I = (0,1). Suppose that is smooth on × I and f is a smooth function on @ × I. Let u(·,�) be the harmonic functions onwith boundary values f(·,�). We show that u(( z,�),�) is smooth on ×I. Our main result is proved for suitable Holder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings ( ·,�) from D, the closure of the unit disc, ontosuch that is smooth on D × I and real analytic at ( √ −1,0) ∈ D × I, but for every family of Riemann mappings R(·,�) fromonto D, the function R(( z,�),�) is not real analytic at ( √ −1,0) ∈ D×I.

/pdf/dirichlet-and-neumann-problems-for-planar-domains-with-1cbgs5mj2h.pdf

Dirichlet and Neumann problems for planar domains with parameter

We construct a finitely dimensional manifold of invariant holomorphic discs attached to a certain class of smooth pseudoconvex hypersurfaces of finite type, generalizing the notion of stationary discs. The discs we construct are determined by a finite jet at a given boundary point and their centers fill an open set. As a consequence, we obtain finite jet determination of CR diffeomorphisms between smooth hypersurfaces of this class.

Florian Bertrand

Papers

Stationary holomorphic discs and finite jet determination problems

Stationary discs and finite jet determination for non-degenerate generic real submanifolds

Dirichlet and Neumann problems for planar domains with parameter

Stationary Discs for Smooth Hypersurfaces of Finite Type and Finite Jet Determination

Sharp estimates of the Kobayashi metric and Gromov hyperbolicity