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

TL;DR

This paper characterizes surfaces with parallel normalized mean curvature vector fields in Euclidean and Minkowski 4-space, introducing canonical parameters and invariant functions to classify and explicitly construct such surfaces.

Contribution

It introduces canonical parameters and invariant functions to describe and classify surfaces with parallel normalized mean curvature vector fields in 4-space, providing a system of PDEs for their characterization.

Findings

Surfaces are uniquely determined by three invariant functions.

Canonical parameters simplify the classification of these surfaces.

Explicit examples of such surfaces are constructed in Euclidean and Minkowski spaces.

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.