• Simonson, Matthew D.
  • Received : 2010.08.19
  • Published : 2011.11.01


We solve the isoperimetric problem, the least-perimeter way to enclose a given area, on various Euclidean, spherical, and hyperbolic surfaces, sometimes with cusps or free boundary. On hyperbolic genus-two surfaces, Adams and Morgan characterized the four possible types of isoperimetric regions. We prove that all four types actually occur and that on every hyperbolic genus-two surface, one of the isoperimetric regions must be an annulus. In a planar annulus bounded by two circles, we show that the leastperimeter way to enclose a given area is an arc against the outer boundary or a pair of spokes. We generalize this result to spherical and hyperbolic surfaces bounded by circles, horocycles, and other constant-curvature curves. In one case the solution alternates back and forth between two types, a phenomenon we have yet to see in the literature. We also examine non-orientable surfaces such as spherical M$\ddot{o}$obius bands and hyperbolic twisted chimney spaces.


isoperimetric problem;hyperbolic surface;mobius band


  1. C. Adams and F. Morgan, Isoperimetric curves on hyperbolic surfaces, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1347-1356.
  2. S. S. Chern, Studies in Global Geometry and Analysis, Englewood Clifts, NJ: Math. Assoc. Am., 1967.
  3. M. Engelstein, A. Marcuccio, Q. Maurmann, and T. Pritchard, Isoperimetric problems on the sphere and on surfaces with density, New York J. Math. 15 (2009), 97-123.
  4. W. Fenchel, Elementary Geometry in Hyperbolic Space, Walter de Gruyter, 1989.
  5. J. Hass and F. Morgan, Geodesics and soap bubbles in surfaces, Math. Z. 223 (1996), no. 2, 185-196.
  6. H. Howards, M. Hutchings, and F. Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999), no. 5, 430-439.
  7. M. Lee, Isoperimetric regions in surfaces and in surfaces with density, Rose-Hulman Und. Math. J. 7 (2006), no. 2, www.rose-
  8. G. Mondello, A criterion for convergence in the augmented Teichmiiuller Space, Bull. Lond. Math. Soc. 41 (2009), 733-746.
  9. F. Morgan, Geometric Measure Theory, A beginner's guide. Fourth edition. Elsevier/Academic Press, Amsterdam, 2009.
  10. P. Schmutz, Riemann surfaces with shortest geodesic of maximal length, Geom. Funct. Anal. 3 (1993), no. 6, 564-631.
  11. M. F. da Silva, Isoperimetric regions in $H^{2}$ between parallel horocycles, (2009), based on doctoral thesis (2006).
  12. W. P. Thurston, The Geometry and Topology of Three-Manifolds, electronic version 1.1, March 2002,
  13. E. W. Weisstein, Double Torus, From MathWorld-A Wolfram Web Resource.
  14. J. Wiegert, The sagacity of circles: a history of the isoperimetric problem, Convergence (2004), Math. Assoc. Amer.