H. Wu constructed a new Hermitian metric that provides an interesting interpolant between the Finsler (like the Carath ́eodory and Kobayashi metrics) and the K ̈ahler invariant metrics. The Wu metric raises several intriguing questions some of which I investigated. One of them is closely related to a problem posed by Kobayashi in 1970: It is well known that if M is a Hermitian manifold with negative holomorphic curvature, then M is Kobayashi hyperbolic. It was natural for Kobayashi to ask whether the converse is also true, i.e., does a Kobayashi hyperbolic manifold M admit a Hermitian metric of negative holomorphic curvature? In a joint project with G. P. Balakumar at ISI Chennai, I studied the Wu metric on convex and non-convex egg domains in Cn. We verified that the Wu metric provides an affirmative answer to the above question of Kobayashi for these egg domains.