JournalsjemsVol. 23, No. 12pp. 3801–3887

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

  • Hajer Bahouri

    CNRS & Sorbonne Université, Paris, France
  • Alaa Marachli

    Université Paris-Est Créteil, France
  • Galina Perelman

    Université Paris-Est Créteil, France
Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to the Simons cone cover
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Abstract

In this article, we establish the existence of a family of hypersurfaces (Γ(t))0<tT(\Gamma(t))_{0< t \leq T} that evolve by the vanishing mean curvature flow in Minkowski space and that, as tt tends to 00, 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)tν+1Q(x/tν+1)u(t,x) \sim t^ {\nu+1} Q( {x} / {t^ {\nu+1}}) , where QQ is a stationary solution and ν\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.

Cite this article

Hajer Bahouri, Alaa Marachli, Galina Perelman, Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to the Simons cone. J. Eur. Math. Soc. 23 (2021), no. 12, pp. 3801–3887

DOI 10.4171/JEMS/1087