Tohoku Mathematical Journal

Real analytic complete non-compact surfaces in Euclidean space with finite total curvature arising as solutions to ODEs

Peter Gilkey, Chan Yong Kim, and JeongHyeong Park

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal upper bound for the total Gauss curvature of the surface which depends only on the orders of the ODEs and we show that the total Gauss curvature of the surface vanishes if the ODEs are second order. We examine when the surfaces are asymptotically minimal.

Article information

Source
Tohoku Math. J. (2), Volume 69, Number 1 (2017), 1-23.

Dates
First available in Project Euclid: 26 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1493172124

Digital Object Identifier
doi:10.2748/tmj/1493172124

Mathematical Reviews number (MathSciNet)
MR3640010

Zentralblatt MATH identifier
1368.53002

Subjects
Primary: 53A05: Surfaces in Euclidean space
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Keywords
Geodesically complete surface finite total Gauss curvature Gauss--Bonnet theorem asymptotically minimal constant coefficient ordinary differential equation

Citation

Gilkey, Peter; Kim, Chan Yong; Park, JeongHyeong. Real analytic complete non-compact surfaces in Euclidean space with finite total curvature arising as solutions to ODEs. Tohoku Math. J. (2) 69 (2017), no. 1, 1--23. doi:10.2748/tmj/1493172124. https://projecteuclid.org/euclid.tmj/1493172124


Export citation

References

  • D. Bleecker, The Gauss–Bonnet Inequality and almost geodesic loops, Adv. in Math. 14 (1974), 183–193.
  • M. Brozos-Vázquez, Universidade da Coruña, Spain – (Mathematica notebook).
  • G. Carron, P. Exner and D. Krejcinik, Topologically nontrivial quantum layers, J. Math. Phys. 45 (2004), 774–784.
  • Q. Chen and Y. Cheng, Chern-Osserman inequality for minimal surfaces in $H^n$, Proc. Amer. Math. Soc. 128 (2000), 2445–2450.
  • S. Chern and R. Osserman, Complete minimal surfaces in euclidean $n$-space, J. Analyse Math. 19 (1967), 15–34.
  • B. Chow, P. Lu and B. Yang, A necessary and sufficient condition for Ricci shrinkers to have positive AVR, Proc. Amer. Math. Soc. 140 (2012), 2179–2181.
  • S. Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math. 2 (1935), 69–133.
  • S. Cohn-Vossen, Totalkrümmung und geodätische linien auf einfach zusammenhän genden offenen volständigen flächenstücken, Recueil Math. Moscow 43 (1936), 139–163.
  • F. Dillen and W. Kühnel, Total curvature of complete submanifolds of Euclidean space, Tohoku Math. J. 57 (2005), 171–200.
  • A. Esteve and V. Palmer, The Chern-Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures, Ark. Mat. 52 (2014), 61–92.
  • P. Gilkey, C. Y. Kim, H. Matsuda, J. H. Park and S. Yorozu, Non-closed curves in $\mathbb{R}^n$ with finite total first curvature arising from the solutions of an ODE, Hokkaido Math. J. 46 (2017).
  • A. Huber, On the subharmonic functions and differential geometry in the large, Comment Math Helv 32 (1957), 13–72.
  • S. Hwang, J. Chang and G. Yun, Variational characterizations of the total scalar curvature and eigenvalues of the Laplacian, Pacific J. Math. 261 (2013), 395–415.
  • L. Jorge and W. Meeks, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), 203–221.
  • M. Kokubu, M. Umehara and K. Yamada, Minimal surfaces that attain equality in the Chern–Osserman inequality, Differential geometry and integrable systems (Tokyo, 2000), 223–228, Contemp. Math., 308, Amer. Math. Soc., Providence, RI, 2002.
  • J. Li, Evolution of eigenvalues along rescaled Ricci flow, Canad. Math. Bull. 56 (2013), 127–135.
  • Y. Li, Y. Luo and H. Tang, On the moving frame of a conformal map from 2-disk into $\mathbb{R}^n$, Calc. Var. Partial Differential Equations 46 (2013), 31–37.
  • X. Ma, Complete stationary surfaces in $\mathbb{R}^4_1$ with total Gaussian curvature $K[M]=-6\pi$, Differential Geom. Appl. 31 (2013),611–622.
  • X. Ma, C. Wang and P. Wang, Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space, Adv. Math. 249 (2013), 311–347.
  • A. Mafra, Finitely curved orbits of complex polynomial vector fields, Anais da Academia Brasileira de Cincias 79 (2007), 13–16.
  • K. Seo, Rigidity of minimal submanifolds in hyperbolic space, Arch. Math. (Basel) 94 (2010), 173–181.
  • K. Shiohama, Cut locus and parallel circles of a closed curve on a Riemannian plane admitting total curvature, Comment. Math. Helv. 60 (1985), 125–138.
  • K. Shiohama, T. Shioya and M. Tanaka, The Geometry of Total Curvature on Complete Open Surfaces, Cambridge Tracts in Mathematics, 159. Cambridge University Press, Cambridge, 2003.
  • K. Shiohama, T. Shioya and M. Tanaka, Mass of rays on complete open surfaces, Pacific J. Math 143 (1990), 349–359.
  • T. Shioya, Behavior of distant maximal geodesics in finitely connected complete 2-dimensional Riemannian manifolds, Mem. Amer. Math. Soc. 108 (1994), no. 517.
  • T. Shioya, Behavior of distant maximal geodesics in finitely connected complete two-dimensional Riemannian manifolds. II, Geom. Dedicata 103 (2004), 1–32.
  • S. Stepanov, I. Tsyganok and J. Mikes, On scalar and total scalar curvatures of Riemann-Cartan manifolds, Kragujevac J. Math. 35 (2011), 291–301.
  • S. Willerton, On the magnitude of spheres, surfaces and other homogeneous spaces, Geom. Dedicata 168 (2014), 291–310.