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

Abstract

We study the Wu metric on convex egg domains of the form E2m = z ∈ C n : |z1| 2m + |z2| 2 + . . . + |zn−1| 2 + |zn| 2 < 1 where m ≥ 1/2, m = 1. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be C 2 -smooth. Overall however, the Wu metric is shown to be continuous when m = 1/2 and even C 1 -smooth for each m > 1/2, and in all cases, a non-Kähler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such E2m.1991 Mathematics Subject Classification. Primary: 32F45; Secondary: 32Q45. Key words and phrases. Wu metric, Kobayashi metric, negative holomorphic curvature. Both the authors were supported by the DST-INSPIRE Fellowship of the Government of India. 1 The term 'holomorphic curvature' stands precisely for the holomorphic sectional curvature and is said to be strongly negative, if it is bounded above by a negative constant.2 We shall reserve the term 'metric' for what was termed as a sub-metric in [13] and use the word 'distance'for what is termed as a 'metric' in general topology and metric geometry. 3 These are also referred to as 'complex ellipsoids' in the complex analysis literature. However, since the ellipsoid in Wu's construction is defined by a polynomial of degree two, we adopt a different terminology.

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

“…In fact, in the case of C 2 -smooth convex eggs, i.e. when m > 1, the Wu metric is only C 1smooth (see [1]) while the Kobayashi metric is C 2 -smooth. For the non-convex pseudo-eggs as in (1.1), first note that ∂E 2m is not even C 1 -smooth.…”

“…We continue our study of the Wu metric from [1] by focusing on the following class of non-convex pseudo-egg domains (1.1)…”