International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 9, Pages 1393-1404

Determination of surfaces in three-dimensional Minkowski and Euclidean spaces based on solutions of the Sinh-Laplace equation

Paul Bracken

Department of Mathematics, University of Texas – Pan American, Edinburg 78541-2999, TX, USA

Received 23 January 2005

Copyright © 2005 Paul Bracken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The relationship between solutions of the sinh-Laplace equation and the determination of various kinds of surfaces of constant Gaussian curvature, both positive and negative, will be investigated here. It is shown that when the metric is given in a particular set of coordinates, the Gaussian curvature is related to the sinh-Laplace equation in a direct way. The fundamental equations of surface theory are found to yield a type of geometrically based Lax pair for the system. Given a particular solution of the sinh-Laplace equation, this Lax can be integrated to determine the three fundamental vectors related to the surface. These are also used to determine the coordinate vector of the surface. Some specific examples of this procedure will be given.