Isoperimetric problems on the sphere and on surfaces with density

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).