A Tale of Two Conformally Invariant Metrics
نویسندگان
چکیده
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.
منابع مشابه
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.
متن کاملOn Positive Solutions to Semi-linear Conformally Invariant Equations on Locally Conformally Flat Manifolds
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn, [g]). We prove that, on a closed and locally conformally flat manifold with...
متن کاملThe conformally invariant Laplace-Beltrami operator and factor ordering
In quantum mechanics the kinetic energy term for a single particle is usually written in the form of the Laplace-Beltrami operator. This operator is a factor ordering of the classical kinetic energy. We investigate other relatively simple factor orderings and show that the only other solution for a conformally flat metric is the conformally invariant Laplace-Beltrami operator. For non-conformal...
متن کاملScattering and Complete Integrability in Conformally Invariant Nonlinear Theories
We study conformally invariant nonlinear wave equations in four dimensions corresponding to multicomponent massless scalar elds with a quartic interaction. We prove that the scattering operator S on the space H of nite-Einstein-energy Cauchy data has innnitely many xed points, as well as periodic points of all orders. There are also 2 H such that S n is almost periodic but not periodic, and 2 H...
متن کاملConserved masses in GHS Einstein and string black holes and consistent thermodynamics.
We analyze the relationship between quasilocal masses calculated for solutions of conformally related theories. We show that the adm mass of a static, spherically symmetric solution is conformally invariant (up to a constant factor) only if the background action functional is conformally invariant. Thus, the requirement of conformal invariance places restrictions on the choice of reference spac...
متن کامل