arXiv subjects

Robert Osserman

Publications and source records attributed to Robert Osserman.

3 recordsLinked to original sources

arXiv1998

A new variant of the Schwarz-Pick-Ahlfors lemma

We prove a ``general shrinking lemma'' that resembles the Schwarz--Pick--Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that {\em all} distances are shrunk. The method of proof is also different in that it relates the shrinking of the Schwarz--Pick--Ahlfors-type lemmas to the comparison theorems of Riemannian geometry.

math.CV↗

arXiv1997

A sharp Schwarz inequality on the boundary

A number of classical results reflect the fact that if a holomorphic function maps the unit disk into itself taking the origin into the origin, and if some boundary point $b$ maps to the boundary, then the map is a magnification at $b$. We prove a sharp quantitative version of this result which also sharpens a classical result of Loewner, and which implies that the map is a strict magnification at $b$ unless it is a rotation.

math.CV↗

arXiv1996

An estimate for the Gauss curvature of minimal surfaces in $\mathbb R^m$ whose Gauss map omits a set of hyperplanes

We give an estimate of the Gauss curvature for minimal surfaces in ${\mathbb R}^m$ whose Gauss map omits more than $m(m+1)/2$ hyperplanes in ${\mathbb P}^{m-1}({\mathbb C})$.

math.DG↗