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

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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

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

First available in Project Euclid: 26 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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