On asymptotic behavior of the mass of rays
HTML articles powered by AMS MathViewer
- by Takashi Shioya PDF
- Proc. Amer. Math. Soc. 108 (1990), 495-505 Request permission
Abstract:
We consider the measure of the set of all unit vectors tangent to rays emanating from a point $p$ in a finitely connected complete open Riemannian $2$-manifold $M$. If $M$ with one end admits total curvature $c(M)$, then this measure tends to $\min \{ 2\pi \chi (M) - c(M),2\pi \}$ as $p$ tends to infinity, where $\chi (M)$ is the Euler characteristic.References
-
S. Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Composito Math. 2 (1935), 63-133.
—, Totalkrümmung und geodätische Linien auf einfach zusammenhängenden offenen volständigen Flächenstücken, Recueil Math. Moscow 43 (1936), 139-163.
- 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
- Philip Hartman, Geodesic parallel coordinates in the large, Amer. J. Math. 86 (1964), 705–727. MR 173222, DOI 10.2307/2373154
- Masao Maeda, On the existence of the rays, Sci. Rep. Yokohama Nat. Univ. Sect. I 26 (1979), 1–4. MR 553892
- Masao Maeda, A geometric significance of total curvature on complete open surfaces, Geometry of geodesics and related topics (Tokyo, 1982) Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1984, pp. 451–458. MR 758663, DOI 10.2969/aspm/00310451
- Masao Maeda, On the total curvature of noncompact Riemannian manifolds. II, Yokohama Math. J. 33 (1985), no. 1-2, 93–101. MR 817975
- Takayuki Oguchi, Total curvature and measure of rays, Proc. Fac. Sci. Tokai Univ. 21 (1986), 1–4. MR 844742
- Koichi Shiga, On a relation between the total curvature and the measure of rays, Tsukuba J. Math. 6 (1982), no. 1, 41–50. MR 675599, DOI 10.21099/tkbjm/1496159448
- Koichi Shiga, A relation between the total curvature and the measure of rays. II, Tohoku Math. J. (2) 36 (1984), no. 1, 149–157. MR 733625, DOI 10.2748/tmj/1178228909
- 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
- Katsuhiro Shiohama, Total curvatures and minimal areas of complete surfaces, Proc. Amer. Math. Soc. 94 (1985), no. 2, 310–316. MR 784184, DOI 10.1090/S0002-9939-1985-0784184-3
- Katsuhiro Shiohama, An integral formula for the measure of rays on complete open surfaces, J. Differential Geom. 23 (1986), no. 2, 197–205. MR 845705 K. Shiohama, T. Shioya and M. Tanaka, Mass of rays on complete open surfaces, preprint.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 495-505
- MSC: Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-1990-0986652-X
- MathSciNet review: 986652