Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

SMYTH SURFACES AND THE DREHRISS

John M. Burnsa1 and Michael J. Clancya2

a1 Department of Mathematics, National University of Ireland Galway, University Road, Galway, Ireland (john.burns@nuigalway.ie)

a2 School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland (michael.clancy@dcu.ie)

Abstract

Isometric deformations of immersed surfaces in Euclidean 3-space are studied by means of the drehriss. When the immersion is of constant mean curvature and the deformation preserves the mean curvature, we determine the drehriss explicitly in terms of the immersion and its Gauss map. These methods are applied to obtain an alternative classification of the Smyth surfaces, i.e. constant mean curvature immersions of the plane into Euclidean 3-space which admit the action of $S^1$ as a non-trivial group of internal isometries.

(Received July 07 2004)

Keywords

  • Primary 3A10;
  • surfaces;
  • constant mean curvature;
  • isometric deformations;
  • Delaunay