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 $\mathbb{R}^{1+M}$ with the dimension $2\leq M\leq7$. In this dimension, the stationary solution of timelike extremal hypersurface equation is the solution of classical minimal surface equation, 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. Here we overcome that the vector field method (based on the energy estimate and bootstrap argument) is lose effectiveness due to the lack of time-decay of solution for the linear perturbation equation. Meanwhile, a global well-posed result of linear damped wave with variable time-space coefficients is established. Hence, our result construct a unique global timelike non-small solution near the hyperplane.