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.2000 Mathematics Subject Classification. Primary 53B20, Secondary 53A07, 53A55.

Fundamental Theorems for Timelike Surfaces in the Minkowski 4-Space

“…The functions β 1 and β 2 characterize the class of surfaces with parallel mean curvature vector field. In [13], we proved that:…”

“…For the class of minimal surfaces in R 4 , the number of the invariant functions and the number of the differential equations determining the surfaces are reduced to two [2]. The surfaces with parallel normalized mean curvature vector field are determined uniquely up to a motion by three invariant functions satisfying a system of three partial differential equations [3].…”

“…Preliminaries. Let R 4 1 be the four-dimensional Minkowski space endowed with the standard flat metric ⟨•,•⟩ of signature (3,1) given in local coordinates by…”

“…The pseudo-orthonormal frame field {x, y, n 1 , n 2 } is geometrically determined: x, y are the two lightlike directions in the tangent space (determined up to notation); n 1 is the unit normal vector field collinear with the mean curvature vector field H; n 2 is determined by the condition that {n 1 , n 2 } is an orthonormal frame field of the normal bundle (n 2 is determined up to a sign). We call this pseudo-orthonormal frame field {x, y, n 1 , n 2 } a geometric frame field of the surface [13].…”