Timelike Meridian Surfaces of Elliptic type in the Minkowski 4-Space
Victoria Bencheva, Velichka Milousheva
TL;DR
The paper investigates timelike meridian surfaces of elliptic type in the Minkowski 4-space $\mathbb{R}^4_1$, focusing on four principal classifications: constant Gauss curvature, constant mean curvature, parallel mean curvature vector field, and parallel normalized mean curvature vector field. It constructs the surfaces as 2D timelike submanifolds lying on rotational hypersurfaces with timelike axis, derives their fundamental forms, and expresses key invariants such as the Gauss curvature $K$, normal curvature $K^{\perp}$, and the mean curvature vector $H$. For each class, complete characterizations are obtained, including explicit meridian data and differential relations (e.g., $\ddot{f}-K f=0$, minimal $H=0$ conditions, and conditions for parallelism of $H$ or $H_0$). The work further derives canonical isotropic parameters, provides the corresponding geometric frame, and uses the results on PNMCVF surfaces to produce explicit solutions to the associated background PDEs, complemented by concrete examples that demonstrate the construction and solvability of these equations. The findings advance Lund-Regge-type analyses in pseudo-Euclidean 4-space and yield explicit PNMCVF solutions, enriching the differential-geometric understanding of timelike surfaces in $\mathbb{R}^4_1$ with potential applications in geometric analysis and mathematical physics.
Abstract
We consider a special family of 2-dimensional timelike surfaces in the Minkowski 4-space $\mathbb{R}^4_1$ which lie on rotational hypersurfaces with timelike axis and call them meridian surfaces of elliptic type. We study the following basic classes of timelike meridian surfaces of elliptic type: with constant Gauss curvature, with constant mean curvature, with parallel mean curvature vector field, with parallel normalized mean curvature vector field. The results obtained for the last class are used to give explicit solutions to the background systems of natural PDEs describing the timelike surfaces with parallel normalized mean curvature vector field in $\mathbb{R}^4_1$.