## Tohoku Mathematical Journal

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

#### 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

https://projecteuclid.org/euclid.tmj/1493172124

Digital Object Identifier
doi:10.2748/tmj/1493172124

Mathematical Reviews number (MathSciNet)
MR3640010

Zentralblatt MATH identifier
1368.53002

#### 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

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