THE ISOPERIMETRIC PROBLEM ON EUCLIDEAN, SPHERICAL, AND HYPERBOLIC SURFACES

- Journal title : Journal of the Korean Mathematical Society
- Volume 48, Issue 6, 2011, pp.1285-1325
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2011.48.6.1285

Title & Authors

THE ISOPERIMETRIC PROBLEM ON EUCLIDEAN, SPHERICAL, AND HYPERBOLIC SURFACES

Simonson, Matthew D.;

Simonson, Matthew D.;

Abstract

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 Mobius bands and hyperbolic twisted chimney spaces.

Keywords

isoperimetric problem;hyperbolic surface;mobius band;

Language

English

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