Some geometric consequences of conformal structure

Abstract

1. Any oriented surface immersed smoothly in E3 may be viewed as a Riemann surface, inheriting the usual conformal structure from the Euclidean metric of the surrounding space. But a strictly convex surface, oriented so that its mean curvature is positive, has still another conformal structure imposed upon it in a natural way. It is the conformal structure obtained by using the second fundamental form as metric tensor. Only on spherical portions of a surface do the two conformal structures coincide. In a recent paper [5], we described geometrically cases in which certain standard differential geometric correspondences are Teichmiiller mappings. There, of course, we worked with the conventional conformal structure on the surfaces involved. But, standard differential geometric correspondences might in certain cases be Teichmiiller mappings between the Riemann surfaces determined by using the second conformal structure described above. And, in still other cases, these correspondences might be Teichmiiller mappings involving the usual conformal structure on one surface, and the second conformal structure on another. In this paper we describe geometrically cases in which such Teichmuiller mappings actually are obtained. Of special interest, perhaps, are the particular instances in which these mappings are conformal. Our results tend to parallel rather closely those obtained in [5]. Wherever possible, lemmas and theorems below have been numbered so as to indicate their correspondence to related items in that previous paper.