Conformally Invariant Metrics and Prime Ends

A Tale of Two Conformally Invariant Metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.

Is mass conformally invariant ?

By using the Garfinkle, Horowitz and Strominger black hole solutions as examples, we illustrate that, with respect to the reference action functional proposed by Hawking and Horowitz, the asymp-totic mass parameter is not invariant between two conformally related static spherically symmetric metrics.

Minimization of Conformally Invariant Energies

Conformally Invariant Non-local Operators

On a conformal manifold with boundary, we construct conformally invariant local boundary conditions B for the conformally invariant power of the Laplacian k , with the property that ( k , B) is formally self-adjoint. These boundary problems are used to construct conformally invariant nonlocal operators on the boundary Σ, generalizing the conformal Dirichlet-to-Robin operator, with principal par...

Conformally Invariant Off - shell Strings ∗

Recent advances in non-critical string theory allow a unique continuation of critical Polyakov string amplitudes to off-shell momenta, while preserving conformal invari-ance. These continuations possess unusual, apparently stringy, characteristics, as we illustrate with our results for three-point functions. 1. The Polyakov functional integral The Polyakov path integral 1 appears only a recipe ...