Abstract. We show that, on holomorphic manifolds that have a plurisubharmonic exhaustion function and that do not carry nonconstant bounded plurisubharmonic functions (e.g. C n ), holomorphic vector fields that are complete in positive time are complete in complex time.
Abstract. Let Σ be a bordered Riemann surface with genus g and m boundary components. Let {γ z } z∈∂Σ be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Let p ∈ Σ and let F be the family of all bounded holomorphic functions f on Σ such that f (p) ≥ 0 and f (z) ∈ γ z for almost every z ∈ ∂Σ. Then there exists a smooth up to the boundary holomorphic function f 0 ∈ F with at most 2g + m − 1 zeros on Σ so that f 0 (z) ∈ γ z for every z ∈ ∂Σ and such that f 0 (p) ≥ f (p) for every f ∈ F. If, in addition, all the curves {γ z } z∈∂Σ are strictly convex, then f 0 is unique among all the functions from the family F .
We consider the existence of solutions of a nonlinear Riemann-Hilbert problem for a quasilinear ∂-equation on a bordered Riemann surface. (2000): Primary 35Q15, 30E25; Secondary 30G20 on such that u 0 (z) belongs to the bounded component of C\γ z for every z ∈ ∂ . Let K be a compact subset of and let ε > 0. Then there exists a smooth up to the boundary solution u of the equation (1.1) on such that u(z) ∈ γ z for every z ∈ ∂ and u − u 0 ∞ < ε on K.
Mathematics Subject Classification
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