Surfaces with Parallel Normalized Mean Curvature Vector Field in Euclidean or Minkowski 4-Space

Abstract

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface  we introduce special isothermal  parameters (canonical parameters) and describe  these surfaces in terms of three invariant functions.
 We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is  determined uniquely up to a motion in Euclidean (or  Minkowski) space by the three invariant functions  satisfying a system of three partial differential equations.  We find examples of surfaces with parallel normalized mean curvature vector field and solutions to the corresponding  systems of PDEs in  Euclidean  or Minkowski space  in the class of the meridian surfaces.