[Paper Review] The relative extremal function for Borel sets in complex manifolds

Armen Edigarian, Ragnar Sigurðsson|ArXiv.org|Jul 13, 2006

Geometry and complex manifolds9 references3 citations

TL;DR

This paper establishes a disc formula for the relative extremal function on Borel sets in complex manifolds, proving that for locally pluriregular Borel sets $ A $ in a weakly regular domain $ D $, the relative extremal function $ \omega(\cdot, \overline{A}, D) $ is bounded above by the infimum of $ -\sigma_f(A) $ over analytic discs $ f $ with $ f(0) = x $, extending Poletsky's theorem to non-open sets. The key contribution is a characterization of pluripolar sets via analytic discs in Josefson manifolds and weakly regular domains.

ABSTRACT

We study a disc formula for the relative extremal function for Borel sets in complex manifolds.

Motivation & Objective

• To extend Poletsky's disc formula for the relative extremal function from open sets to Borel sets in complex manifolds.
• To investigate the validity of the inequality $ \omega(\cdot, \overline{A}, X) \leq \Omega(\cdot, \overline{A}, X) $ for Borel sets $ A $ in complex manifolds.
• To characterize pluripolar sets in terms of analytic discs in weakly regular domains and Josefson manifolds.
• To determine conditions under which the relative extremal function for a Borel set $ A \subset \partial D $ satisfies $ \omega(\cdot, \overline{A}, D) \leq \Omega(\cdot, \overline{A}, D) \leq \omega^{*}(\cdot, A, D) $.

Proposed method

• Define the relative extremal function $ \omega(\cdot, A, X) $ as the supremum of non-positive plurisubharmonic functions bounded above by $ -\chi_A $.
• Introduce $ \Omega(x, A, X) $ as the infimum of $ -\sigma_f(A) $ over analytic discs $ f \in \mathcal{O}(\mathbb{D}, X) \cap C(\overline{\mathbb{D}}, X) $ with $ f(0) = x $.
• Use the subaverage property of plurisubharmonic functions to derive $ \omega(x, A, X) \leq \Omega(x, A, X) $ for all Borel sets $ A $.
• Prove that for locally pluriregular $ A $, $ \Omega(\cdot, \overline{A}, X) \leq \omega(\cdot, A, X) $ in a complex manifold $ X $.
• Establish that in a relatively compact weakly regular domain $ D $ of a Josefson manifold, $ \Omega(\cdot, A, D) \leq \omega^{*}(\cdot, A, D) $ for Borel $ A \subset \partial D $.
• Construct analytic discs inductively via a sequence of neighborhoods and use the Poisson integral and Perron-Bremermann envelopes to control boundary behavior.

Experimental results

Research questions

• RQ1 Under what conditions does the disc formula $ \Omega(x, A, X) = \omega(x, A, X) $ hold for Borel sets $ A $ in complex manifolds?
• RQ2 Can the relative extremal function for a Borel set $ A \subset \partial D $ in a weakly regular domain $ D $ be characterized via analytic discs?
• RQ3 Is the inequality $ \Omega(\cdot, \overline{A}, D) \leq \omega^{*}(\cdot, A, D) $ valid for all Borel subsets $ A \subset \partial D $?
• RQ4 What is the relationship between the pluripolarity of a Borel set $ A \subset \partial D $ and the existence of analytic discs with large $ \sigma_f(A) $?
• RQ5 Does the property that every bounded plurisubharmonic function on a Josefson manifold is constant characterize such manifolds via analytic disc approximation?

Key findings

• For any locally pluriregular Borel set $ A $ in a complex manifold $ X $, the inequality $ \Omega(\cdot, \overline{A}, X) \leq \omega(\cdot, A, X) $ holds.
• In a relatively compact weakly regular domain $ D $ of a Josefson manifold, $ \Omega(\cdot, A, D) \leq \omega^{*}(\cdot, A, D) $ for any Borel set $ A \subset \partial D $.
• If $ A = A_1 \cup E $ with $ A_1 $ locally pluriregular and $ E $ such that $ u^*|_E \equiv -\infty $ for some $ u \in \mathrm{PSH}^-(D) $, then $ \omega(x, \overline{A}, D) \leq \Omega(x, \overline{A}, D) \leq \omega^*(x, A, D) $ for all $ x \in D $.
• A Josefson manifold has the property that every bounded plurisubharmonic function is constant if and only if for every $ p \in X $, every non-pluripolar Borel $ A \subset X $, and every $ \varepsilon > 0 $, there exists an analytic disc $ f \in \mathcal{O}(\mathbb{D}, X) \cap C(\overline{\mathbb{D}}, X) $ with $ f(0) = p $ and $ \sigma_f(A) > 1 - \varepsilon $.
• A compact set $ A \subset \partial D $ satisfies $ \omega^*(\cdot, A, D) \equiv 0 $ if and only if there exists $ u \in \mathrm{PSH}^-(D) $, $ u \not\equiv -\infty $, such that $ u^*|_A \equiv -\infty $.