Conformally invariant metrics and uniform structures. I

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.

Se p 20 04 Conformally invariant energies of knots I ∗

Conformally invariant functionals on the space of knots are introduced via extrinsic confor-mal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does,...

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...