Friedmann-Lemaitre-Robertson-Walker时空中给定平均曲率函数的整体类空超曲面研究

本项目拟研究Friedmann-Lemaitre-Robertson-Walker时空中给定平均曲率方程的类空解。具体研究下列问题：FLRW时空中当曲率函数为“凹凸函数之和类型”且允许参数为负值时的类空图的分歧结构；FLRW时空中给定平均曲率方程的类空解的存在性、多解性、解集全局分歧结构以及正解连通分支是一维流形的判据；FLRW时空中类空解的渐近行为及内部层现象；球形区域上FLRW空间中给定平均曲率方程Dirichlet问题的相应有限差分问题正解的逼近对称性。本研究不仅会促进平均曲率方程的理论发展，而且对类空超曲面的数值求解以及宇宙学具有重要的理论意义。

The proposal will aim to study the entire spacelike hypersurface with prescribed mean curvature function in certain Friedmann-Lemaitre-Robertson-Walker spacetimes. It includes that: the global bifurcation structure of a spacelike graph problem in FLRW spacetimes in which the curvature function is “the sum of a convex function and a concave function” and the parameters are allowed to be negative; the existence and multiplicity of spacelike hypersurface with prescribed mean curvature function in certain FLRW spacetimes；the global bifurcation structure of entire radial solutions and the conditions to guarantee positive solution sets are of one-dimension manifolds; the asymptotic behavior of spacelike solutions and the phenomena of interior layers in FLRW spacetimes; the approximate symmetry theorems for radial spacelike hypersurface of the finite difference problems associated to the mean curvature equations in a ball. The research is expected not only to solve prescribed mean curvature equations in FLRW space, but also to provide theoretical implications for computing the numerical solutions of spacelike hypersurface，and it is of great theoretical significance to cosmology.