The Kobayashi pseudodistance on almost complex manifolds

We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familliar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold and finish with some examples.Comment: 19 pages, Late

Linear Problems For The Schwarzian Derivative (rigid Domain).

In this thesis we consider the set U of Schwarzian derivatives of all univalent holomorphic functions in the unit disk, as a subset of the Banach space of holomorphic functions with finite hyperbolic sup-norm of weight $-2.$ The concept of local extreme point of U is defined, and it is shown, using subordination, that if the Schwarzian derivative S(f) of f is a local extreme point of U, then f cannot omit an open set. For isolated points of U we show that f cannot omit a set of positive measure. This follows from the fact that the complement of an arbitrary rigid domain in the sense of Thurston, has zero measure. The proof uses an existence theorem for holomorphic Lipschitz functions due to Nguyen Xuan Uy. We also consider the support points of U, and show by examples that there are support points that omit open sets, and so U has support points that are not extreme points.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128065/2/8720322.pd

Injective hyperbolicity of domains

Abstract. The pseudometric of Hahn is identical to the Kobayashi-Royden pseudometric on domains of dimension greater than two. Thus injective hyperbolicity coincides with ordinary hyperbolicity in this case. 1. Introduction. The Kobayashi pseudodistance d M and KobayashiRoyden pseudodifferential metric K M of a complex manifold M are defined by means of extremal problems for holomorphic mappings of the unit disk D into M . By restricting to injective holomorphic mappings in these extremal problems, one arrives at a pseudodistance τ M and a pseudodifferential metric S M respectively. These were considered first on plane domains by Si

Pseudoholomorphic mappings and Kobayashi hyperbolicity

We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familiar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold. We give special consideration to almost complex structures tamed by some symplectic form. The notions and pseudoholomorphic curves involved are illustrated in some examples. Introduction The Poincar'e metric on the open unit disk D j in the complex plane C j is a Riemannian metric jvj = jvj euc 1 \Gamma jzj 2 conformal with the Euclidean metric j \Delta j euc , that induces a distance d on D j with the remarkable property that every holomorphic mapping f : D j ! D j is distance nonincreasing in d. This fact, discovered in 1915 by Pick [22] is an invariant formulation of the Schwarz lemma. By means of holomorphic mappings of the unit disk into a complex manifold M , Kobayashi in 1967 used the distance d to define a pseudodistance dM on M . This has the p..