 

Max Engelstein, Anthony Marcuccio,
Quinn Maurmann, and Taryn Pritchard
Isoperimetric problems on the sphere and on surfaces with density


Published: 
March 17, 2009 
Keywords: 
Minimal partitions, isoperimetric problem, surfaces with density, stability 
Subject: 
53C42 


Abstract
We discuss partitions of the sphere and other ellipsoids into equal
areas and isoperimetric problems on surfaces with density. We prove that the
leastperimeter partition of any ellipsoid into two equal areas is by division
along the shortest equator. We extend the work of C. Quinn, 2007,
and give a new sufficient condition for a perimeterminimizing partition of
S^{2} into four regions of equal area to be the tetrahedral arrangement
of geodesic triangles. We solve the isoperimetric problem on the plane with
density y^{α} for α>0 and solve the double bubble problem when
α is a positive integer. We also identify isoperimetric regions on
cylinders with densities e^{z} and θ^{α}. Next, we investigate
stable curves on surfaces of revolution with radially symmetric densities.
Finally, we give an asymptotic estimate for the minimal perimeter of a
partition of any smooth, compact surface with density into n regions of equal
area, generalizing the previous work of Maurmann et al. (to appear).


Author information
Max Engelstein:
Department of Mathematics, Yale University, New Haven, CT 06520
max.engelstein@yale.edu
Anthony Marcuccio:
Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267
08anm@williams.edu
Quinn Maurmann:
Department of Mathematics, Brown University, Providence, RI 02912
quinn_maurmann@brown.edu
Taryn Pritchard:
Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267
08tbp@williams.edu

