Null scrolls as solutions of a sigma model (Q2881347)

The authors give some results on the two-dimensional nonlinear sigma model with boundary. The paper contains the following sections: nonlinear sigma model with boundary, solutions through elastic curves in the 2-sphere, solutions through null scrolls etc. The authors formulate four questions and obtain an ample response to these. Let \(L^3\) be the three-dimensional Lorentz-Minkowski space. The following theorems are obtained:NEWLINENEWLINE A) (i) Every null scroll is a Willmore surface in \(L^3\). (2) Null scrolls provide solutions of the two-dimensional \(O(2,1)\) nonlinear sigma model.NEWLINENEWLINEB) A Lorentzian surface in the Lorentz-Minkowski three space is a null scroll if and only if \(H^2- K=0\). Mean and Gauss curvatures functions of the null scroll are given by the relations: \(H(s,t)= f(s)\), \(K(s,t)= f(s)^2\). The space of field configurations of the two-dimensional \(O(2,1)\) nonlinear sigma model contains certain classes of solutions.NEWLINENEWLINE Examples: a) Rotational surfaces with either time-like or null axis and profile curve being a time-like free elastic curve in an anti-de Sitter plane. b) Right cylinders whose section is a time-like free elastic curve in a Lorentzian plane.