Parabolic Classical Curvature Flows

Authors

BRENDAN GUILFOYLE, School of Science, Technology, Engineering and Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.Follow
Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom.

Document Type

Article

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Disciplines

Applied Mathematics | Other Physical Sciences and Mathematics | Physical Sciences and Mathematics

Publication Details

This article has been published in a revised form in the Journal of the Australian Mathematical Society [https://doi.org/10.1017/S1446788717000210]. This version is published under a Creative Commons CC-BY-NC-ND. No commercial re-distribution or re-use allowed. Derivative works cannot be distributed. © copyright holder.

Abstract

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Recommended Citation

GUILFOYLE, B., & KLINGENBERG, W. (2018). PARABOLIC CLASSICAL CURVATURE FLOWS. Journal of the Australian Mathematical Society, 104(3), 338-357. doi:10.1017/S1446788717000210