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PARABOLIC CLASSICAL CURVATURE FLOWS

Part of: Classical differential geometry Parabolic equations and systems

Published online by Cambridge University Press:  30 October 2017

BRENDAN GUILFOYLE  and
WILHELM KLINGENBERG
BRENDAN GUILFOYLE*
Affiliation:
School of STEM, Institute of Technology, Tralee, Co. Kerry, Ireland email brendan.guilfoyle@ittralee.ie
WILHELM KLINGENBERG
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK email wilhelm.klingenberg@durham.ac.uk
*
✉brendan.guilfoyle@ittralee.ie
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Abstract

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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.

Keywords

oriented linesradii of curvatureparabolic flowWeingartencurvature flow

MSC classification

Primary: 35K40: Second-order parabolic systems
Secondary: 53A05: Surfaces in Euclidean space
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 3 , June 2018 , pp. 338 - 357
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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