Timelike Surfaces with Parallel Normalized Mean Curvature Vector Field in the Minkowski 4-Space
Victoria Bencheva, Velichka Milousheva
TL;DR
This work studies timelike surfaces in the Minkowski 4-space $\mathbb{R}^4_1$ with parallel normalized mean curvature vector field, introducing canonical isotropic parameters to reduce the surface data to three geometric functions $\lambda$, $\mu$, $\nu$. The authors derive a system of natural PDEs that these functions must satisfy, with separate regimes for $K-H^2>0$, $K-H^2<0$, and $K-H^2=0$, thereby solving the Lund-Regge problem for this class. The canonical parametrization yields explicit expressions for the second fundamental form and enables a constructive proof of local existence and uniqueness: any admissible triple $(\lambda,\mu,\nu)$ generates a unique surface up to rigid motion. This framework unifies prior results on parallel mean curvature and its normalized variant in both Euclidean and Minkowski settings, and provides a finite PDE-driven data set to classify and construct such surfaces. The results have potential implications for the geometric analysis of timelike submanifolds and related physical models in general relativity and string theory.
Abstract
In the present paper, we study timelike surfaces with parallel normalized mean curvature vector field in the four-dimensional Minkowski space. We introduce special isotropic parameters on each such surface, which we call canonical parameters, and prove a fundamental existence and uniqueness theorem stating that each timelike surface with parallel normalized mean curvature vector field is determined up to a rigid motion in the Minkowski space by three geometric functions satisfying a system of three partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, thus solving the Lund-Regge problem for this class of surfaces.