How to use MATHEMATICA to find cyclic surfaces of constant curvature in Loretz-Minkowski space
by
Rafael López
Universidad de Granada, Spain

Oral Communication

A cyclic surface in Euclidean three-dimensional space \r³ is a surface foliated by pieces of circles, that is, it is generated by a smooth uniparametric family of pieces of circles. In particular, the surfaces of revolution are cyclic surfaces. In 1785, Meusnier proved that the catenoid is the only minimal rotational surface in \r³. In the XIXth century, Riemann found all complete minimal surfaces foliated by circles in parallel planes. A Riemann surface is a simply periodic embedded surface and it is defined in terms of elliptic functions. At the same time, Enneper showed that the planes of the foliation in a minimal cyclic surface must be parallel. More recently, Nitsche proved in 1989 that the only cyclic surfaces with non-zero constant mean curvature are the surfaces of revolution, that is, the surfaces discovered by Delaunay in 1841.

In Lorentz-Minkowski 3-space \l³, a circle is defined as the orbit of a point p away from a straight-line l under the action of the group of rotations in \l³ that leaves l pointwise fixed. In this note, we study cyclic non-degenerate surfaces in \l³ with constant mean curvature or constant Gauss curvature. The main results can be summarised as follows (see ):

Let M be a (spacelike or timelike) cyclic surface in \l³. Assume the mean curvature or Gauss curvature is constant. Then either the planes containing the circles must be parallel or M is a subset of a pseudohyperbolic surface or a pseudosphere.

When the curvature is a non-zero constant, we obtain:

All (spacelike or timelike) cyclic surfaces in \l³ with non-zero constant mean curvature or non-zero constant Gauss curvature are surfaces of revolution.

In the case that the mean curvature or Gauss curvature vanishes on the surface, we shall give a complete description of them. The explicit parametrisations of these surfaces involve elliptic functions, as it occurs in \r³ with the minimal surfaces of Riemann type. It should be pointed out that the major goal in this note is the discovery of a family of non-rotational maximal spacelike surfaces in \l³ foliated by circles in parallel planes that play, in some sense, the same role as Riemann surfaces in \r³. A drawing of a maximal surface in \l³ of Riemann type can be seen at the URL http://www.ugr.es/local/rcamino/riemann1.gif. Exactly, we prove

There exists a family of maximal spacelike surfaces foliated by circles in parallel spacelike, timelike or lightlike planes and invariant under a group of translations.

A sketch of the proofs is as follows. The reasoning in Theorem 1 is local and by contradiction. If the planes of the foliation are not parallel, consider an unit vector field orthogonal to each of these planes and and let an integral curve c of this vector field. We consider an appropriate parametrisation of the surface in terms of the Frenet frame of c. The computation of the mean curvature or Gauss curvature yields a polynomical equation in several variables. The fact that the coefficients of this polynomial vanish gets a contradiction. Assuming now that the planes of the foliation are parallel, the proof of Theorem 2 is a new computation of the curvature, but with an easier parametrisation. This shall conclude that the surface must be rotational.

In proofs of our results, the software system Mathematica plays an important role in a twofold sense. First, we need to realise an explicit computation of the curvature for a surface in \l³. This requires a tedious symbolic calculation to do by hand and of which Mathematica program simplifies enormously the work. A second application of Mathematica in our researh consists on obtaining graphical plots of the new Riemann surfaces discovered in \l³. These surfaces are usually defined in terms of elliptic functions. The special library of Mathematica solves numerically the calculations, producing interesting drawings for visualizing these surfaces. These images allows us a description of their geometrical properties such as the periodicity, embeddeness and existence of ends and cone type points.

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F. J. López, R. López, R. Souam, Maximal surfaces of Riemann type in Lorentz-Minkowski space \l³, preprint.

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R. López, Constant mean curvature hypersurfaces foliated by spheres, Diff. Geom. Appl. 11 (1999), 245-256.

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R. López, Constant mean curvature surfaces foliated by circles in Lorentz-Minkowski space, Geom. Dedicata, 76 (1999), 81-95.

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R. López, Timelike surfaces in Lorentz 3-space with constant mean curvature, to appear in Tohoku Math. J.

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R. López, Cyclic hypersurfaces of constant curvature, to appear in Advances Studies in Pure Mathematics.

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R. López, Cyclic surfaces of constant Gauss curvature, preprint.

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R. López, Surfaces of constant Gauss curvature in Lorentz-Minkowski 3-space, preprint.

http://www.ugr.es/local/rcamino

Date received: January 27, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-09.