Rotational Surfaces in $${\mathbb{L}^3}$$ and Solutions of the Nonlinear Sigma Model

Abstract

The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2, 1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational symmetry is completely determined. Essentially, the solutions are warped products of orbits of the 1-dimensional groups of isometries and elastic curves in either a de Sitter plane, a hyperbolic plane or an anti de Sitter plane. The main tools are the equivalence of the two-dimensional O(2, 1) Nonlinear Sigma Model and the Willmore problem, and the description of the surfaces with rotational symmetry. A complete classification of such surfaces is obtained in this paper. Indeed, a huge new family of Lorentzian rotational surfaces with a space-like axis is presented. The description of this new class of surfaces is based on a technique of surgery and a gluing process, which is illustrated by an algorithm. MSC 2000 Classification: Primary 53C40; Secondary 53C50

Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space

“…(1) First, the surface may be considered as the graph of a function defined in each one of the coordinate planes of R 3 , indicating that the problem depends on the choice of this plane. (2) We may also assume that this plane is lightlike since the coordinate planes are not lightlike. because the domain of the integrand function is x 2 > 1 and the domain of arc tanh(x) is x 2 < 1.…”

Willmore-Like Tori in Killing Submersions

“…Now, the following lemma collects some formulae that will be needed in the sequel. They can be obtained using similar computations to those included in [13,22]. For simplicity, we have eliminated the symbol in most of our notation.…”

“…Finally, we combine (20) and (22) to characterize the extremals of W Φ ( ) as the solutions of the following Euler-Lagrange equation:…”

“…In contrast to the case of Willmore surfaces in ambient spaces of constant curvature, there are not many results concerning Chen-Willmore submanifolds in background spaces with nonconstant sectional curvature. Nevertheless, Willmore surfaces and submanifolds have strong connections in Physics with applications to (just to mention a few) the analysis of elastic plates and biological membranes [1,20] and to bosonic string theories and sigma models (for more details, see [21,22] and references therein). Several of these applications are based on a beautiful link between Willmore surfaces and elastica which can be established by using a symmetry reduction procedure [12].…”

On Angles and Pseudo-Angles in Minkowskian Planes

“…For related literature on geometry and its applications, see, e.g., also [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”