Determination of surfaces in three‐dimensional Minkowski andEuclidean spaces based on solutions of the Sinh‐Laplace equation | Zendy

Paul Bracken | Zendy

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Determination of surfaces in three‐dimensional Minkowski andEuclidean spaces based on solutions of the Sinh‐Laplace equation

Author(s) -

Paul Bracken

Publication year - 2005

Publication title -

international journal of mathematics and mathematical sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.21

H-Index - 39

eISSN - 1687-0425

pISSN - 0161-1712

DOI - 10.1155/ijmms.2005.1393

Subject(s) - mathematics , gaussian curvature , laplace's equation , mathematical analysis , laplace transform , minkowski space , euclidean geometry , surface (topology) , curvature , hyperbolic function , mean curvature , constant curvature , geometry , partial differential equation

The relationship between solutions of the sinh-Laplaceequation and the determination of various kinds ofsurfaces of constant Gaussian curvature, both positiveand negative, will be investigated here. It is shownthat when the metric is given in a particular setof coordinates, the Gaussian curvature is related tothe sinh-Laplace equation in a direct way. Thefundamental equations of surface theory are found toyield a type of geometrically based Lax pair for the system.Given a particular solution of the sinh-Laplaceequation, this Lax can be integrated to determinethe 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

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