Parabolic Classical Curvature Flows

Guilfoyle, B.; Klingenberg, W.

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.

Citation

Guilfoyle, B., & Klingenberg, W. (2018). Parabolic Classical Curvature Flows. Journal of the Australian Mathematical Society, 104(3), 338-357. https://doi.org/10.1017/s1446788717000210

• Journal Article Type: Article
• Acceptance Date: Jan 29, 2017
• Online Publication Date: Oct 30, 2017
• Publication Date: Jun 1, 2018
• Deposit Date: Jan 26, 2017
• Publicly Available Date: Apr 30, 2018
• Journal: Journal of the Australian Mathematical Society
• Print ISSN: 1446-7887
• Electronic ISSN: 1446-8107
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 104
• Issue: 3
• Pages: 338-357
• DOI: https://doi.org/10.1017/s1446788717000210
• Public URL: https://durham-repository.worktribe.com/output/1395685
• Related Public URLs: https://arxiv.org/abs/1503.01930