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Asymptotics of Perimeter-Minimizing Partitions

Published:2010-05-11
Printed: Sep 2010
• Quinn Maurmann,
Department of Mathematics, UCLA, Los Angeles, CA, USA
• Max Engelstein,
Department of Mathematics, Yale University, New Haven, CT, USA
• Anthony Marcuccio,
Department of Mathematics and Statistics, Williams College, Williamstown, MA, USA
• Taryn Pritchard,
Department of Mathematics and Statistics, Williams College, Williamstown, MA, USA
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Abstract

We prove that the least perimeter $P(n)$ of a partition of a smooth, compact Riemannian surface into $n$ regions of equal area $A$ is asymptotic to $n/2$ times the perimeter of a planar regular hexagon of area $A$. Along the way, we derive tighter estimates for flat tori, Klein bottles, truncated cylinders, and Möbius bands.
 MSC Classifications: 53C42 - Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]