Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

Černe, Miran

doi:10.1007/bf02384500

Cited by 5 publications

(2 citation statements)

References 22 publications

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“…There is also a strong connection of the solutions of the Riemann-Hilbert problem to the description of the polynomial hull of a compact set K fibered over ∂∆, [1,7,8,16,23,24,27,28,29,35,36,37,38]. Recall that the polynomial hull K of a compact set K ⊆ C n is defined as…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

Černe¹

2004

ajm

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Section: Introductionmentioning

confidence: 99%

Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

Černe¹

2004

ajm

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“…A circular hypersurface (that is, a hypersurface invariant under rotations ↦ ) is strongly ℂ-convex if and only if it is strongly convex [Čer,Prop. 3.7].…”

Section: Projective Dual Structuresmentioning

confidence: 99%

Sums of CR functions from competing CR structures

Barrett

Grundmeier

2018

Pacific J. Math.

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In this paper we characterize sums of CR functions from competing CR structures in two scenarios. In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions.

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Polynomial hulls and an optimization problem

Whittlesey

2004

J Geom Anal

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Abstract. We say that a subset of C n is hypoconvex if its complement is the union of complex hyperplanes. We say it is strictly hypoconvex if it is smoothly bounded hypoconvex and at every point of the boundary the real Hessian of its defining function is positive definite on the complex tangent space at that point. Let B n be the open unit ball in C n . Suppose K is a C ∞ compact manifold in ∂B 1 × C n , n > 1, diffeomorphic to ∂B 1 × ∂B n , each of whose fibers K z over ∂B 1 bounds a strictly hypoconvex connected open set. Let K be the polynomial hull of K. Then we show that K \ K is the union of graphs of analytic vector valued functions on B 1 . This result shows that an unnatural assumption regarding the deformability of K in an earlier version of this result is unnecessary. Next, we study an H ∞ optimization problem. If ρ is a C ∞ real-valued function on ∂B 1 × C n , we show that the infimum γ ρ = inf f ∈H ∞ (B 1 ) n ρ(z, f (z)) ∞ is attained by a unique bounded f provided that the set {(z, w) ∈ ∂B 1 × C n |ρ(z, w) ≤ γ ρ } has bounded connected strictly hypoconvex fibers over the circle. §1 Introduction and results. The purpose of this work is to strengthen the results of [Wh] and [V] regarding the existence of analytic structure in a class of polynomial hulls and the solution of an H ∞ optimization problem.For w ∈ C n let w j denote the j th coordinate of w and let |w| denote the standard Euclidean norm. If Y is a compact set in C n , then the polynomial (convex) hull Y of Y is given by Y = {z ∈ C n |P (z)| ≤ sup w∈Y |P (w)| for all polynomials P on C n }.

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Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

Cited by 5 publications

References 22 publications

Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

Sums of CR functions from competing CR structures

Polynomial hulls and an optimization problem

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