## Abstract

We discuss partit ions of the sphere and other ellipsoids into equal areas and isoperimetric problems on surfaces with density. We prove that the least-perimeter 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 isoperimetrie 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).

Original language | English (US) |
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Pages (from-to) | 97-123 |

Number of pages | 27 |

Journal | New York Journal of Mathematics |

Volume | 15 |

State | Published - 2009 |

Externally published | Yes |

## Keywords

- Isoperimetric problem
- Minimal partitions
- Stability
- Surfaces with density