THE ISOPERIMETRIC PROBLEM ON EUCLIDEAN, SPHERICAL, AND HYPERBOLIC SURFACES
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...
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Veröffentlicht in: | Journal of the Korean Mathematical Society 2011, Vol.48 (6), p.1285-1325 |
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Format: | Artikel |
Sprache: | kor |
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Zusammenfassung: | 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. |
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ISSN: | 0304-9914 |