We begin by studying characterizations of pseudoconvexity, and also of finite type, using analytic discs. The results presented are analogous to well-known ideas from the real variable setting in which “pseudoconvex” is replaced by “convex” and “analytic disc” is replaced by “line segment”. The second part of the paper concerns regularity results for the Kobayashi metric. Of course this metric ...

We show that if a locally compact group G acts properly on a locally compact σ-compact space X, then there is a family of G-invariant proper continuous finite-valued pseudometrics which induces the topology of X. If X is, furthermore, metrizable, then G acts properly on X if and only if there exists a G-invariant proper compatible metric on X.

We show a surgery formula for the relative Yamabe invariant and give applications to the study of concordance classes of metrics.

The Eguchi–Hanson, Taub–NUT and Atiyah–Hitchin metrics are the only complete non–singular SO(3)–invariant hyper–Kahler metrics in four dimensions. The presence of a rotational SO(2) isometry allows for their unified treatment based on solutions of the 3–dim continual Toda equation. We determine the Toda potential in each case and examine the free field realization of the corresponding solutions...

In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient condition that left-invariant Randers metrics are of Berwald type is given. Fin...

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.

Nature reveals itself in similar structures of different scales. A child and an adult share similar organs yet dramatically differ in size. Comparing the two is a challenging task to a computerized approach as scale and shape are coupled. Recently, it was shown that a local measure based on the Gaussian curvature can be used to normalize the local metric of a surface and then to extract global ...

The unit ball B is shown to be Gromov hyperbolic with respect to the Ferrand metric λBn and the modulus metric μBn , and dimension dependent upper bounds for the Gromov delta are obtained. In the two-dimensional case Gromov hyperbolicity is proved for all simply connected domains G. For λG also the case G = R n \ {0} is studied.

Expressions are derived for the normalized root-mean-square error of an image relative to a reference image. Different versions of the error metric are invariant to different combinations of effects, including the image's (a) being multiplied by a real or complex-valued constant, (b) having a constant added to its phase, (c) being translated, or (d) being complex conjugated and rotated 180 degr...