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Cyclic pursuit on compact manifolds

Gekhtman, Dmitri (2017) Cyclic pursuit on compact manifolds. Pacific Journal of Mathematics, 289 (1). pp. 153-168. ISSN 0030-8730. doi:10.2140/pjm.2017.289.153.

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We study a form of cyclic pursuit on Riemannian manifolds with positive injectivity radius. We conjecture that on a compact manifold, the piecewise geodesic loop formed by connecting consecutive pursuit agents either collapses to a point in finite time or converges to a closed geodesic. The main result is that this conjecture is valid for nonpositively curved compact manifolds.

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Additional Information:© 2017 Mathematical Sciences Publishers. Received: 4 October 2016; Revised: 18 December 2016; Accepted: 29 December 2016; Published: 12 May 2017. This research was conducted mostly at the SUMMER@ICERM Undergraduate Summer Research Program in 2012. I would like to thank Tarik Aougab and Sergei Tabachnikov for their mentorship. I would like to thank Francisc Bozgan for pointing out the application of Jensen’s inequality in Section 3. I would like to thank Anton Petrunin for a MathOverflow answer which helped with the proof of Proposition 8.1.
Subject Keywords:cyclic pursuit, curve shortening, closed geodesics, nonpositive curvature
Issue or Number:1
Classification Code:MSC2010: primary 53B21; secondary 37D40
Record Number:CaltechAUTHORS:20170712-131427876
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79015
Deposited By: Tony Diaz
Deposited On:12 Jul 2017 21:25
Last Modified:15 Nov 2021 17:44

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