论文标题
无效锥体中的等距嵌入问题
The Isometric Embedding Problem in a Null Cone
论文作者
论文摘要
在本文的第一部分中，我们将线性嵌入问题的Li-Wang的结果扩展到了任意维度的紧凑型歧管。然后，我们表明，与环境几何形状无关，嵌入式$ n $ sphere的任何度量扰动也均匀地嵌入到同质codazzi方程的溶液中。在第二部分中，我们专门研究尺寸二，并在环境空锥中研究这些结果。具体而言，给定在2个球体上的指标路径和初始等轴测嵌入，我们为等距嵌入的路径开发了一个小的参数存在和唯一定理。在最后一部分中，在对零锥上施加渐近衰减条件后，我们表明，在2个球体上的任何度量都可以嵌入到缩放因子上。然后，我们证明了在无限邻里的叶面存在。
In the first part of this paper, we extend the result of Li-Wang on the linearized embedding problem to a compact manifold of arbitrary dimension. Using this, we then show that any metric perturbation of a embedded $n$-sphere is also isometrically embedded up to a solution of the homogenous Codazzi equation, irrespective of the ambient geometry. In the second part we specialize to dimension two, and study these results within an ambient Null Cone. Specifically, given a path of metrics on the 2-sphere and an initial isometric embedding, we develop a small parameter existence and uniqueness theorem for paths of isometric embeddings. In the final part, after imposing asymptotic decay conditions on the Null Cone, we show that any metric on the 2-sphere can be isometrically embedded up to a scaling factor. We then prove the existence of a foliation in a neighborhood of infinity.
