Total curvatures and minimal areas of complete surfaces

Shiohama, Katsuhiro

doi:10.1090/S0002-9939-1985-0784184-3

Total curvatures and minimal areas of complete surfaces
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by Katsuhiro Shiohama

Proc. Amer. Math. Soc. 94 (1985), 310-316

DOI: https://doi.org/10.1090/S0002-9939-1985-0784184-3

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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 $4\pi$.

References

Stefan Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math. 2 (1935), 69–133 (German). MR 1556908
F. Fiala, Le problème des isopérimètres sur les surfaces ouvertes à courbure positive, Comment. Math. Helv. 13 (1941), 293–346 (French). MR 6422, DOI 10.1007/BF01378068
Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
Philip Hartman, Geodesic parallel coordinates in the large, Amer. J. Math. 86 (1964), 705–727. MR 173222, DOI 10.2307/2373154
Alfred Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13–72. MR 94452, DOI 10.1007/BF02564570
Katsuhiro Shiohama, The role of total curvature on complete noncompact Riemannian $2$-manifolds, Illinois J. Math. 28 (1984), no. 4, 597–620. MR 761993
Katsuhiro Shiohama, Cut locus and parallel circles of a closed curve on a Riemannian plane admitting total curvature, Comment. Math. Helv. 60 (1985), no. 1, 125–138. MR 787666, DOI 10.1007/BF02567404