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

TL;DR

This paper constructs complete, non-compact surfaces in Euclidean space with finite total curvature using solutions of second-order ODEs, providing bounds and conditions for minimality and embedding properties.

Contribution

It introduces a method to generate such surfaces from ODE solutions and characterizes their geometric properties, including curvature bounds and minimality conditions.

Findings

Universal upper bound for total Gauss curvature depending on ODE order

Total Gauss curvature vanishes for second-order ODEs

Surfaces can be asymptotically minimal under certain conditions

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.