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Centro-affine curvature flows on centrally symmetric convex curves


Author: Mohammad N. Ivaki
Journal: Trans. Amer. Math. Soc. 366 (2014), 5671-5692
MSC (2010): Primary 53C44, 53A04, 52A10, 53A15; Secondary 35K55
DOI: https://doi.org/10.1090/S0002-9947-2014-05928-X
Published electronically: July 21, 2014
MathSciNet review: 3256179
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Abstract: We consider two types of $ p$-centro-affine flows on smooth, centrally symmetric, closed convex planar curves: $ p$-contracting and $ p$-expanding. Here $ p$ is an arbitrary real number greater than $ 1$. We show that, under any $ p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $ \pi $ converge, in the Hausdorff metric, to the unit circle modulo $ SL(2)$. As a $ p$-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any $ p$-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized to enclose constant area $ \pi $, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to $ SL(2)$ transformations, to the unit circle. At the end of the paper, we present a new proof of the $ p$-affine isoperimetric inequality, $ p\geq 1$, for smooth, centrally symmetric convex bodies in $ \mathbb{R}^2$.


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Additional Information

Mohammad N. Ivaki
Affiliation: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada H3G 1M8
Email: mivaki@mathstat.concordia.ca

DOI: https://doi.org/10.1090/S0002-9947-2014-05928-X
Received by editor(s): September 17, 2011
Received by editor(s) in revised form: June 11, 2012
Published electronically: July 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.