Conformal Capacities and Extremal Metrics

For any non-compact Riemannian manifold M of dimension n ≥ 2 we previously defined a function λM : M×M → R+ = R+∪{+∞] only dependent on the conformal structure of M , and proved that for a class of manifolds containing all the proper subdomains of R, λ 1 n M was a distance on M [F1, F2]. The case of a domain G of R has been the object of several investigations leading to estimations of λG[V1, . . . , V4] and to properties of the λG-Lipschitz mappings [FMV]. Then by considering the case of a ball M. Vuorinen has been led to conjecture that λ 1 1−n G is also a distance. In some way this improvement of my previous result is the best possible as it was proved in [AVV] that λ−p G cannot be a metric if p > 1 n−1 . Moreover its interest is reinforced by the fact that, if n = 2, it is relevant to the Teichmuller’s theory of quadratic differentials and to the Jenkins’s extremal metrics [J1]. This conjecture has been proven in [AVV] for G = B, by A.Y.Solynin [S] and J.A. Jenkins [J2] for G = R \ {0}, then extended by J.A. Jenkins [J2] to any plane domain with finite connectivity. But this extension is somewhat difficult to follow as it involves some families of homotopy groups which are not easy to define with precision when G is not simply-connected. In the present paper we will prove the following general result, which includes the Vuorinen conjecture: