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

## TL;DR

This paper constructs hypersurfaces evolving under vanishing mean curvature flow in Minkowski space that develop singularities resembling the Simons cone, using a novel approach for a quasilinear wave equation.

## Contribution

It establishes the existence of finite-time blow-up solutions for a quasilinear wave equation modeling hyperbolic vanishing mean curvature flow with singularity formation.

## Key findings

- Existence of hypersurfaces blowing up towards Simons cone
- Construction of solutions with specific asymptotic behavior
- Handling of quasilinear difficulties in wave equation analysis

## Abstract

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.