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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Total curvatures and minimal areas of complete surfaces

Author: Katsuhiro Shiohama
Journal: Proc. Amer. Math. Soc. 94 (1985), 310-316
MSC: Primary 53C20
MathSciNet review: 784184
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Abstract: Minimal areas for certain classes of finitely connected complete open surfaces are obtained by using a Bonnesen-style isoperimetric inequality for large balls on the surfaces. In particular, the minimal area of Riemannian planes whose Gaussian curvatures are bounded above by 1 is $ 4\pi $.

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  • [1] S. Cohn-Vossen, Kürzeste Wege and Totalkrümmung auf Flächen, Compositio Math. 2 (1935) 69-133. MR 1556908
  • [2] F. Fiala, Le problème isopérimétres sur les surfaces ouvertes à courbure positive, Comment. Math. Helv. 13 (1940), 293-346. MR 0006422 (3:301b)
  • [3] M. Gromov, Volume and bounded cohomology, Publ. Math. I.H.E.S. 56 (1982), 5-100. MR 686042 (84h:53053)
  • [4] P. Hartman, Geodesic parallel coordinates in the large, Amer. J. Math. 86 (1964), 705-727. MR 0173222 (30:3435)
  • [5] A. Huber, On subharmonic functions and differential geometry, in the large, Comment. Math. Helv. 32 (1957), 13-72. MR 0094452 (20:970)
  • [6] K. Shiohama, A role of total curvature on complete noncompact Riemannian $ 2$-manifolds, Illinois J. Math. 28 (1984), 597-620. MR 761993 (86h:53069)
  • [7] -, Cut locus and parallel circles of a closed curve on a Riemannian plane admitting total curvature, Comment. Math. Helv. (to appear). MR 787666 (86g:53049)

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Keywords: Complete manifolds, Gaussian curvature, geodesics, isoperimetric inequality
Article copyright: © Copyright 1985 American Mathematical Society

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