1 O ct 1 99 3 Complex Finsler metrics by Marco Abate and

A complex Finsler metric is an upper semicontinuous function F : T 1,0 M → R + defined on the holomorphic tangent bundle of a complex Finsler manifold M , with the property that F (p; ζv) = |ζ|F (p; v) for any (p; v) ∈ T 1,0 M and ζ ∈ C. Complex Finsler metrics do occur naturally in function theory of several variables. The Kobayashi metric introduced in 1967 ([K1]) and its companion the Carathéodory metric are remarkable examples which have become standard tools for anybody working in complex analysis; we refer the reader to [K2, 4], [L], [A] and [JP] to get an idea of the amazing developments in this area achieved in the past 25 years. In general, the Kobayashi metric is not at all regular; it may even not be continuous. But in 1981 Lempert [Le] proved that the Kobayashi metric of a bounded strongly convex domain D in C n is smooth (outside the zero section of T 1,0 D), thus allowing in principle the use of differential geometric techniques in the study of function theory over strongly convex domains (see also Pang [P2] for other examples of domains with smooth Kobayashi metric). We started dealing with this kind of problems in [AP1]. In particular, [AP2] was devoted to the search of differential geometric conditions ensuring the existence in a complex Finsler manifold of a foliation in holomorphic disks like the one found by Lempert in strongly convex domains, where the disks were isometric embeddings of the unit disk ∆ ⊂ C endowed with the Poincaré metric. And indeed (see also [AP3]) we found necessary and sufficient conditions (see also Pang [P1] for closely related results). In that case, because the nature of the problem required the solution of certain P.D.E.'s, the conditions were mainly expressed in local coordinates somewhat hiding their geometric meaning. The aim of this paper is to present an introduction to complex Finsler geometry in a way suitable to deal with global questions. Roughly speaking, the idea is to isometrically embed a complex Finsler manifold into a hermitian vector bundle, and then apply standard hermitian differential geometry techniques, in the spirit of [K3]. Here we provide just a coarse outline of the procedure. Let˜M be the complement of the zero section in T 1,0 M. We assume that the complex Finsler metric F is smooth oñ M , and that …