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)|
|Number of pages||27|
|Journal||New York Journal of Mathematics|
|State||Published - 2009|
- Isoperimetric problem
- Minimal partitions
- Surfaces with density