Communications in Analysis and Geometry
Volume 28 (2020)
The Second of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday
Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau
On isolated umbilic points
Pages: 2005 – 2018
Counter-examples to the famous conjecture of Carathéodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean $3$-space.
In particular, Riemannian metrics with a smooth strictly convex $2$-sphere containing a single umbilic point are constructed explicitly, in contradiction with any direct extension of Carathéodory’s conjecture to non-Euclidean metrics. Additionally, a non-Euclidean metric and embedded surfaces containing isolated umbilic points of any index are presented, violating Hamburger’s umbilic indexbound.
In both cases, it is shown that the metric can be made arbitrarily close to the flat metric.
Received 2 December 2018
Accepted 11 October 2020
Published 8 January 2021