Evolution of spacelike surfaces in $$\mathrm{AdS}_3$$ by their Lagrangian angle

Abstract

We study spacelike hypersurfaces \(M\) in an anti-De Sitter spacetime \(N\) of constant sectional curvature \(-\kappa , \kappa >0\) that evolve by the Lagrangian angle of their Gaus maps. In the two dimensional case we prove a convergence result to a maximal spacelike surface, if the Gaus curvature \(K\) of the initial surface \(M\subset N\) and the sectional curvature of \(N\) satisfy \(|K|<\kappa \).