Isometric deformation of surfaces in R 3 preserving the mean curvature function

Abstract

The purpose of this paper is to classify surfaces in Euclidean 3- space with constant Gaussian curvature which admit non-trivial oneparameter families of isometric immersions preserving the mean curvature function. It is shown that the Gaussian curvature must be zero and, if the mean curvature is not constant, then such isometric immersions are some deformations of the cylinder over a logarithmic spiral.