On the upper semicontinuity of the Wu metric

Abstract

Abstract. We discuss continuity and upper semicontinuity of the Wu pseudometric.The Wu pseudometric was introduced by H. Wu in [Wu 1993] (and [Wu]). Various properties of the Wu metric may be found for instance in [Che-Kim 1996], [Che-Kim 1997], [Kim 1998], [Che-Kim 2003], [Juc 2002]. Nevertheless, it seems that even quite elementary properties of this metric are not completely understood, e.g. its upper semicontinuity.First, let us formulate the definition of the Wu pseudometric in an abstract setting. Let h : C n −→ R + be a C-seminorm. Put:= the orthogonal complement of V with respect to the standard Hermitian scalar product z, w :Consider the family F of all pseudo-Hermitian scalar products s :where s 0 := s| U×U (note that E(s 0 ) = E(s) ∩ U ). Let Vol(s 0 ) denote the volume of E(s 0 ) with respect to the Lebesgue measure of U . Since I 0 is bounded, there exists an s ∈ F with Vol(s 0 ) < +∞. Observe that for any basis e = (e 1 , . . . , e m ) of U (m := dim C U ) we have

Analysing the Wu metric on a class of eggs in ℂ n – II

“…While the Wu metric is indeed a norm, it is not clear if the Wu metric enjoys better regularity than the Kobayashi metric. In general, the Wu metric may fail to be upper semicontinuous [8] notwithstanding the fact that the Kobayashi metric is always upper semicontinuous. In fact, in the case of C 2 -smooth convex eggs, i.e.…”

“…, z n ) of E 2m . Indeed, the continuity and completeness of the Wu metric on E 2m follows from Proposition 4 of [8]. Completeness of the Wu metric here relies on Kobayashi completeness of the domains E 2m , which is guaranteed by [11].…”

Analysing the Wu metric on a class of eggs in ℂ n – I

“…Being functorial and the simplest modified (Hermitian) form of the Kobayashi metric, the Wu metric is the first natural candidate to be investigated for (K-W). However, the Wu metric may fail to be upper semicontinuous [8], notwithstanding the fact that the Kobayashi metric is always upper semicontinuous. But then again, the Wu metric seems to reflect the boundary geometry better in special instances.…”