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

Abstract

We study the Wu metric for the non-convex domains of the formwhere 0 < m < 1/2. Explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs E2m are obtained. The Wu metric is then verified to be a continuous Hermitian metric on E2m which is real analytic everywhere except along the complex hypersurface Z = {(0, z2, . . . , zn) ∈ E2m}. We also show that the holomorphic sectional curvature of the Wu metric for this non-compact family of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of m. This verifies a conjecture of S. Kobayashi and H. Wu for such E2m.1991 Mathematics Subject Classification. Primary: 32F45; Secondary: 32Q45, 32H15.

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

“…Using Theorem 3.1, one can also verify that the holomorphic curvature of the Wu metric is bounded between a pair of negative constants on the domain U Z = E 2m \Z. The strong negativity of the holomorphic curvature current across the remaining thin set Z (surrounded by U Z ) follows using arguments similar to those outlined in Section 5 (for 1/2 < m < 1) and Section 3 of [2] (when m = 1/2). Remark 3.2.…”

“…The purpose of this note and its counterpart [2], is to carry forward the case study of the Wu metric in conjunction with (K-W) as in [5] and [6], to egg domains of the form (1.1)…”

“…By Lempert's work [11], all invariant metrics on convex bounded domains, which turn holomorphic maps into Lipschitz 1-maps, coincide with the Kobayashi metric. Analysis of the non-convex eggs is dealt with in a separate article [2], for clarity.…”

“…Any invariant metric, in particular the Wu and the Kobayashi metric, can be written down explicitly on E 2m once it is known at points of S ∪{0} where {p ∈ C n : p = (p 1 , 0), 0 < p 1 < 1}. Indeed, the automorphism 2 and Ψ is the automorphism of B n−1 taking p to the origin, given by…”