Einstein双曲几何流方程Cauchy问题的整体解

The Global Solutions to Cauchy Problem in Einstein Hyperbolic Geometric Flow Equation

本文中我们研究了一类定义在n维黎曼流形上的双曲几何流,并证明了小初值情况下的Cauchy问题的光滑解随时间整体存在。当变量t趋向于无穷时,由几何流方程的解所决定的度量的数量曲率趋向于0。该结论表明这类特殊的黎曼度量可以流成平坦度量。换而言之,具有适当初始度量的黎曼流形在双曲几何流下演化成欧氏空间。

In this paper, we study the hyperbolic geometric flow which defined on the n-dimensional Riemannian manifold, and proved the global smooth solution of the small initial value of the Cauchy problem. When it goes to the infinity, the scalar curvature determined by the geometric equation goes to 0. This conclusion shows that this special Riemannian metric evolved to the flat metric. In other words, If we choose the properly initial metric for the Riemannian manifold, it will evolve to the Euclidean space.