2019/07/02 by Weiping Yan, Yan, Weiping
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Geometric Analysis and Curvature Flows #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.1907.01126
This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime ℝ1+3. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime ℝ1+3, the geometry of them are two spheres. The linear mode unstable of those lightlike self-similar solutions for the radially symmetric membranes equation is given. After that, we show those self-similar solutions of the radially symmetric membranes equation are nonlinearly stable inside a strictly proper subset of the backward lightcone. This means that the dynamical behavior of those two spheres is as attractors. Meanwhile, we overcome the double roots case (the theorem of Poincaré can't be used) in solving the difference equation by construction of a Newton's polygon when we carry out the analysis of spectrum for the linear operator.