Tohoku Mathematical Journal
2017
March
SECOND SERIES VOL. 69, NO. 1

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

Title REAL ANALYTIC COMPLETE NON-COMPACT SURFACES IN EUCLIDEAN SPACE WITH FINITE TOTAL CURVATURE ARISING AS SOLUTIONS TO ODES

Author Peter Gilkey, Chan Yong Kim and JeongHyeong Park

(Received November 7, 2014, revised April 20, 2015)
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.

Mathematics Subject Classification. Primary 53A05; Secondary 53C21.
Key words and phrases. Geodesically complete surface, finite total Gauss curvature, Gauss--Bonnet theorem, asymptotically minimal, constant coefficient ordinary differential equation.

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