J. Korean Math. Soc. 2011; 48(6): 1285-1325
Printed November 1, 2011
https://doi.org/10.4134/JKMS.2011.48.6.1285
Copyright © The Korean Mathematical Society.
Matthew D. Simonson
Milton Academy
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 least-perimeter 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\"obius bands and hyperbolic twisted chimney spaces.
Keywords: isoperimetric problem, hyperbolic surface, mobius band
MSC numbers: 53C99, 49Q99, 53A99, 30F99
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