Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to the Simons cone

Abstract

In this article, we establish the existence of a family of hypersurfaces $(\Gamma (t){)}_{0<t\le T}$ that evolve by the vanishing mean curvature flow in Minkowski space and that, as $t$ tends to $0$, blow up towards a hypersurface that behaves like the Simons cone both near the origin and at infinity. This issue amounts to singularity formation for a second-order quasilinear wave equation. Our constructive approach consists in proving the existence of finite-time blow up solutions of this hyperbolic equation of the form $u(t,x)\sim t^{\nu +1}Q(x/t^{\nu +1})$ , where $Q$ is a stationary solution and $\nu$ an arbitrarily large positive irrational number. Our approach roughly follows that of Krieger, Schlag and Tataru [22–24]. However, in contrast to these works, the equation to be handled in this article is quasilinear. This brings about a number of difficulties to overcome.