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

Abstract

In this paper, we establish the existence of a family of surfaces (Γ(t))0<t≤T that evolve by the vanishing mean curvature flow in Minkowski space and, as t tends to 0, blow up towards a surface that behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second-order quasilinear wave equation. Our constructive approach consists in proving the existence of a finite-time blow-up solution to this hyperbolic equation under the form u(t,x)∼tν+1Q(xtν+1), where Q is a stationary solution and ν is an irrational number strictly larger than 1/2. Our strategy roughly follows that of Krieger, Schlag and Tataru in [7–9]. However, contrary to these articles, the equation to be handled in this work is quasilinear. This induces a number of difficulties to face.