This paper studies the asymptotic behaviour of Amsler surfaces, i.e., surfaces in \(\mathbb{R}^3\) of negative constant Gauss curvature, which contain two asymptotic lines which are straight lines in \(\mathbb{R}^3\). It is well known that the differential equation, which characetrizes the angle \(\Phi\) between asymptotic lines of surfaces of constant Gauss curvature, is the sine-Gordon equation \({\Phi}_{xy} = ab\sin{\Phi}\). The crucial observation for this paper is (Lemma 3): The angle function \(\Phi(x,y)\) of an Amsler surface is a similarity solution \(\Phi(x,y) = \phi(r)\), where \(r = \sqrt{-4abxy}\), of the sine-Gordon equation.
Thus \(\phi\) is a solution to the third Painleve equation. As a consequence one obtains (Theorem 2 and Corollary 3): An Amsler surface is uniquely characterized by the angle \(\phi(0)\) between its straight asymptotic lines, where \(0<\phi(0) \leq {\pi/2}\).
In particular, the associated family of an Amsler surface consists of one surface. The basis for the explicit asymptotics presented in this paper is the very explicit form of the three partial differential equations describing the frame “moving” in the variables \(x,y\) and \(\lambda\), where \(\lambda\) is the parameter defining the associated family. An evaluation of these three equations leads to a fairly special parametrization of the Amsler surface (Proposition 2). An asymptotic development of the various components involved in this special parametrization (Theorem 3) finally leads (in view of Corollary 5 and Theorem 4) to a fairly explicit asymptotic cone, independent of \(\lambda\) for the Amsler surface. It is shown next that a certain surface of constant mean curvature (Smyth surface) has the same asymptotic cone, proving a conjecture of Pinkall. Finally, two generating curves for the asymptotic cone are given. One satisfies the smoke-ring-propagation equation, and the other one has a very simple curvature formula.