Isoperimetric problems on the sphere and on surfaces with density

Max Engelstein, Anthony Marcuccio, Quinn Maurmann, Taryn Pritchard

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 languageEnglish (US)
Pages (from-to)97-123
Number of pages27
JournalNew York Journal of Mathematics
StatePublished - 2009
Externally publishedYes


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


Dive into the research topics of 'Isoperimetric problems on the sphere and on surfaces with density'. Together they form a unique fingerprint.

Cite this