Global Stability Dynamics of the Timelike Extremal Hypersurfaces in Minkowski Space

This paper aims to study the relationship between the timelike extremal hypersurfaces and the classical minimal surfaces. This target also gives the long time dynamics of timelike extremal hypersurfaces in Minkowski spacetime \(\mathbb {R}^{1+M}\) ${\mathbb{R}}^{1+M}$ with the dimension \(M\ge 2\) $M\ge 2$ . In this dimension, the stationary solution of timelike extremal hypersurface equation is the solution of classical minimal surface equation \(\begin{aligned} \textrm{div}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =0, \quad \forall x\in \mathbb {R}^M, \end{aligned}\) $$\begin{array}{c}\hfill \text{div}\left(\frac{\mathrm{\nabla }u}{\sqrt{1+{|\mathrm{\nabla }u|}^2}}\right)=0,\forall x\in {\mathbb{R}}^M,\end{array}$$ which only admits the hyperplane solution by Bernstein theorem. We prove that this hyperplane solution as the stationary solution of timelike extremal hypersurface equation is asymptotic stablely by finding the hidden dissipative structure of linearized equation. Our result construct a unique global timelike non-small solution near the hyperplane.