In this article, we establish the existence of a family of hypersurfaces $(\Gamma (t))_{0< t \leq T}$ which evolve by the vanishing mean curvature flow in Minkowski space and which as $t$ tends to~$0$ blow up towards a hypersurface which 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 finite time blow up solutions of this hyperbolic equation under the form $u(t,x) \sim t^ {\nu+1} Q\Big(\frac {x} {t^ {\nu+1}} \Big) $, where~$Q$ is a stationary solution and $\nu$ an arbitrary large positive irrational number. Our approach roughly follows that of Krieger, Schlag and Tataru. However contrary to these works, the equation to be handled in this article is quasilinear. This induces a number of difficulties to face.

中文翻译：

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渐近于 Simons 锥的曲面的双曲线消失平均曲率流的爆破动力学

在本文中，我们建立了超曲面族 $(\Gamma (t))_{0< t \leq T}$ 的存在性，它们通过 Minkowski 空间中的平均曲率流消失而演化，并且随着 $t$ 趋向于~$0$ 向一个超曲面爆炸，它的行为类似于无穷远处的 Simons 锥。这个问题相当于研究二阶拟线性波动方程的奇点形成。我们的建设性方法包括证明该双曲方程的有限时间膨胀解的存在形式为 $u(t,x) \sim t^ {\nu+1} Q\Big(\frac {x} {t^ {\nu+1}} \Big) $，其中~$Q$ 是一个平稳解，$\nu$ 是一个任意大的正无理数。我们的方法大致遵循 Krieger、Schlag 和 Tataru 的方法。然而与这些工作相反，本文中要处理的方程是拟线性的。