周期KdV方程的圆、格拉斯曼和几何上的测地流
Geodesic Flows on Diffeomorphisms of the Circle, Grassmannians, and the Geometry of the Periodic KdV Equation
摘要：
首先，我们构造一个希尔伯特流形T的方向保持的圆（模的双全纯自映射组的圆盘）的自同态。这个空间，可以被认为是泛Teichmueller空间的完备化，被赋予了一个右不变的Kaehler度量。利用拟共形映射理论的结果，我们构造了T到无穷维Segal-Wilson Grassmannian的一个嵌入。对于T来说，后者是一个非常自然的环境空间。这使我们能够证明T的截面曲率在全纯方向上是负的，并且沿着Cartan-Hadamard理论的路线进行推理，其测地线存在于所有时间。T的测地线通过V. Arnold对欧拉方程的推广导出了周期KdV方程的解。作为应用，我们得到了具有初始数据的周期KdV方程在指数为3/2的周期Sobolev空间的某个闭子空间中解的长期存在性。
英文摘要：
We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal Teichmueller space, is endowed with a right-invariant Kaehler metric. Using results from the theory of quasiconformal mappings we construct an embedding of T into the infinite dimensional Segal-Wilson Grassmannian. The latter turns out to be a very natural ambient space for T. This allows us to prove that T's sectional curvature is negative in the holomorphic directions and by a reasoning along the lines of Cartan-Hadamard's theory that its geodesics exist for all time. The geodesics of T lead to solutions of the periodic Korteweg-de Vries (KdV) equation by means of V. Arnold's generalization of Euler's equation. As an application, we obtain long-time existence of solutions to the periodic KdV equation with initial data in a certain closed subspace of the periodic Sobolev space of index 3/2.
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