Complex analytic and differential geometry by Demailly J.-P.

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By Demailly J.-P.

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G. ψ(ζ, z) = max{|ζ|2 + | Re z|2 , − log δΩ (ζ, z)}. There is slowly increasing sequence Cj → +∞ such that each function ψj = (Cj − ψ ⋆ ρ1/j )−1 is an “exhaustion” of a pseudoconvex open set Ωj ⊂⊂ Ω whose slices are convex tubes and such that d(Ωj , ∁Ω) > 2/j. Then 1 vj (ζ, z) = v ⋆ ρ1/j (ζ, z) + | Re z|2 + ψj (ζ, z) j is a decreasing sequence of plurisubharmonic functions on Ωj satisfying our previous conditions. As v = lim vj , we see that u = lim uj is plurisubharmonic. 6) Corollary. Let Ω ⊂ Cp × Cn be a pseudoconvex open set such that all slices Ωζ , ζ ∈ Cp , are convex tubes in Cn .

If R < 1, we get a contradiction as follows. Let ψ ∈ Psh(Ω) be an exhaustion function and K = h ∂D(1) × D(R) ⊂⊂ Ω, c = sup ψ. K As ψ ◦ h is plurisubharmonic on a neighborhood of D(1) × D(R), the maximum principle applied with respect to t implies ψ ◦ h(t, w) c on D(1) × D(R), hence h D(1) × D(R) ⊂ Ωc ⊂⊂ Ω and h D(1) × D(R + ε) ⊂ Ω for some ε > 0, a contradiction. d) =⇒ e). The function − log d(z, ∁Ω) is continuous on Ω and satisfies the mean value inequality because − log d(z, ∁Ω) = sup − log δΩ (z, ξ) .

Convexity Properties The close analogy of plurisubharmonicity with the concept of convexity strongly suggests that there are deeper connections between these notions. We describe here a few elementary facts illustrating this philosophy. B (Kiselman’s minimum principle). 13) Theorem. If Ω = ω + iω ′ where ω, ω ′ are open subsets of Rn , and if u(z) is a plurisubharmonic function on Ω that depends only on x = Re z, then ω ∋ x −→ u(x) is convex. Proof. This is clear when u ∈ C 2 (Ω, R), for ∂ 2 u/∂zj ∂z k = 41 ∂ 2 u/∂xj ∂xk .

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