# CR Analytic Varieties with Given Boundary

• Pierre Dolbeault
Part of the The University Series in Mathematics book series (USMA)

## Abstract

Let us recall the Hartogs-Bochner-Martinelli theorem: let Ω be a bounded open set in ℂ n (n ≥ 2), with smooth connected boundary ∂Ω, and let f be a smooth CR function on ∂Ω. Then there exists FO(Ω) smooth on Ω̄ such that F | ∂Ω = f. The graph of f , gr f is a maximally complex submanifold of ℂ n+1 of dimension 2n — 1 ; gr(F | Ω) is a complex submanifold in ℂ n+1 \gr f of complex dimension n, and the boundary of gr̄ is gr f This gives a solution of a boundary problem in ℂ n+1 . We consider such boundary problems, with the solution given by geometric measure theory, so singularities of small measure will be allowed. Moreover, it will be convenient to consider linear combinations of subvarieties.

## Keywords

Compact Support Boundary Problem Simple Extension Geometric Measure Theory Complex Submanifold

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