Ahlfors Theorems for Differential Forms

Some counterparts of theorems of Phragmén-Lindel？f and of Ahlfors are proved for differential forms of -classes. 1. -Forms This paper is continuation of the earlier work [1], where the main topic was to examine the connection between quasiregular (qr) mappings and so-called -classes of differential forms. We first recall some basic notation and terminology from [1]. Let be a Riemannian manifold of class , , with or without boundary, and let be a weakly closed differential form on , that is, for each form with a compact in and such that ; we have Here , , and is the orthogonal complement of a differential form on a Riemannian manifold . A weakly closed form of the kind (1.1) is said to be of the class on if there exists a weakly closed differential form such that almost everywhere on we have for some constant . The differential form (1.1) is said to be of the class on if there exists a differential form (1.4) such that almost everywhere on for some constants . Theorem 1.1. For a proof see [1]. The following partial integration formula for differential forms is useful [1]. Lemma 1.2. Let and be differential forms, , , , , and let have a compact support . Then In particular, the form is weakly closed if and only if a.e. on . Let and be Riemannian manifolds of dimensions , , , and with scalar products , , respectively. The Cartesian product has the natural structure of a Riemannian manifold with the scalar product We denote by and the natural projections of the manifold onto submanifolds. If and are volume forms on and , respectively, then the differential form is a volume form on . Let be an orthonormal system of coordinates in , . Let be a domain in , and let be an -dimensional Riemannian manifold. We consider the manifold . 2. Boundary Sets Below we introduce the notions of parabolic and hyperbolic type of boundary sets on noncompact Riemannian manifolds and study exhaustion functions of such sets. We also present some illuminating examples. Let be an -dimensional noncompact Riemannian manifold without boundary. Boundary sets on are analogies to prime ends due to Carathéodory (cf. e.g., [2]). Let , be a collection of open sets with the following properties:(i)for all ,？？ ,(ii) . A sequence with these properties will be called a chain on the manifold . Let , be two chains of open sets on . We will say that the chain is contained in the chain , if for each there exists a number such that for all we have . Two chains, each of which is contained in the other one, are called equivalent. Each equivalence class of chains is called a boundary set of the manifold