Global stability dynamics of the timelike extremal hypersurfaces in Minkowski space

TL;DR

This paper investigates the long-term behavior of timelike extremal hypersurfaces in Minkowski space, proving the asymptotic stability of hyperplane solutions through novel analysis of linearized equations and global well-posedness results.

Contribution

It establishes the asymptotic stability of hyperplane solutions for timelike extremal hypersurfaces in Minkowski space by uncovering hidden dissipative structures and developing new analytical techniques.

Findings

Hyperplane solutions are asymptotically stable in the studied dimension range.

A global well-posedness result for linear damped wave equations with variable coefficients.

Construction of a unique global timelike non-small solution near the hyperplane.

Abstract

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.