대한수학회지 (Journal of the Korean Mathematical Society)

  • 제48권6호
  • /
  • Pages.1285-1325
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  • 2011
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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THE ISOPERIMETRIC PROBLEM ON EUCLIDEAN, SPHERICAL, AND HYPERBOLIC SURFACES

  • Simonson, Matthew D. (Department of Mathematics Milton Academy)
  • 투고 : 2010.08.19
  • 발행 : 2011.11.01
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초록

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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참고문헌

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