Real analytic complete non-compact surfaces in Euclidean space with finite total curvature arising as solutions to odes (Q2396260)

In this interesting paper, the authors investigate a family of non-compact real analytic isometric embeddings \(\Sigma\) of the plane in \(n\)-dimensional Euclidean space which arise as the solution space to a pair of ODE's. Under some natural conditions on the associated characteristic polynomials, the authors prove that the surface \(\Sigma\) is properly embedded, is geodesically complete and has infinite volume. Moreover, it is also shown that the total Gauss curvature \(K[\Sigma]\) is well defined and, using the Gauss-Bonnet theorem, the authors express \(K[\Sigma]\) in terms of integrals along the coordinates curves. A uniform estimate for \(K[\Sigma]\) is also given. The paper ends with a lot of examples illustrating the main results, most of them being Mathematica assisted.