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
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.