A MODULUS AND AN EXTREMAL FORM OF A FOLIATION

Abstract

We dene a p-modulus of a subset of leaves of a foliation on Riemannian manifold. We prove, that the p-modulus of the foliation is conformal invariant and study the problem of existing of the extremal form for a foliaton on Riemannian manifold. We also compute the value of p-modulus and the extremal form for k-dimensional foliation given by a submersion. The idea of modulus is directly connected with the concept of extremal length of curves in 2 introduced by Beurling and Ahlfors [AhBe] in the beginning of 50-ties. In 1957 Fuglede generelized this notion to the modulus of k-dimensional surface families in n . It was very usefull tool in the theory of conformal and quasiconformal maps, extremely popular in 60-ties and 70-ties. Using a geometric characterization Suominen [Su] extended the modulus to the case of an arbitrary dierential Riemannian manifold, and in 1979 Krivov [Kr] dened generalized p-modulus for a family of k-forms. The modulus of a foliation, introduced by the author in [Bl], connects Fuglede’s and Krivov’s ideas. We used the fact that the foliation of Riemannian manifold may be dened as a family of surfaces or by a family of forms. The modulus of the foliation is just a modulus in Krivov’s sence of the family of forms characteristic for the foliation. For these forms, by Hodge star, arieses the family of dual forms. Both these classes seem to characterize pairs of foliations orthogonal to each other, but this is an open problem yet.