On biholomorphic continuability of regular quasiconformal mappings

Abstract

Almost nothing is known about biholomorphic and even holomorphic continuability of quasiconformal mappings into complex manifolds whose real dimension is twice the dimension of the original manifold. The same applies to biholomorphic continuability of plane quasiconformal mappings into ∉². The problem is very important, e.g. in various approaches to the physics of elementary particles involving the complex geometry of the natural world.

Quasiconformal mappings may be treated as homeomorphic solutions of certain systems of homogenous elliptic partial differential equations which in the two-dimensional case reduce to the Beltrami differential equation, whereas in higher dimensions are nonlinear. Therefore we prefer to utilize here the geometrical variational techniques. The modern geometrical variational method of extremal length is due to Ahlfors and Beurling. On complex manifolds some analogues have been developed by Kobayashi and by Chern, Levine and Nirenberg. For hermitian manifolds it has been introduced by the present author in the Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 839–844, and then developed in later papers. It has proved to be quite useful for the problem in question.

The inverse problem of quasiconformality of projections of biholomorphic mappings was studied in the same Bulletin 23 (1975), 845–851. Here, we firstly extend some results of this paper and make a suitable restriction for further purposes: we distinguish the class of mappings transforming infinitesimal spheres onto (infinitesimal) circular ellipsoids. The extensions obtained still depend on one order of quasiconformality, denoted here by Q. We show, however, that in order to formulate the problem of biholomorphic continuability we need a dependence on two orders of quasiconformality, denoted by Q and Q'. Finally it appears that there are three types of continuability theorems.