Lorentz流形中的类时极值超曲面理论是涉及到几何学、物理学和流体动力学等领域的交叉研究课题，是当前主流的研究方向之一。它在广义相对论、宇宙学和现代数学中具有重要的科学意义和应用背景。本项目将采用双曲几何分析的方法着重研究类时极值超曲面方程在(1) 小扰动Minkowski时空；(2) Schwarzschild时空；(3) 膨胀宇宙时空中柯西问题光滑解的整体存在性。这些问题是连接双曲偏微分方程与几何、物理的一座桥梁，它们的解决无论在理论上还是在应用方面均具有重要的科学价值。

The theory of time-like extremal hypersurfaces in Lorentz manifolds tightly relates to the geometry、physics and fluid dynamics, and is one of the hot research topics nowadays. It has important scientific significance and wide background in general relativity、cosmology and modern mathematics. This project will focus on the global existence of the smooth solution to the Cauchy problem of time-like extremal hypersurface equation in Lorentz manifolds including (1) Minkowski spacetime with small perturbations; (2) Schwarzschild spacetime; (3) expanding spacetime by adopting the method of hyperbolic geometry analysis. These problems build a bridge between hyperbolic partial differential equations and geometry and physics, the resolution of them plays an important role not only in the theory but also in the applications.