二维德西特空间上曲率流的孤子解及其应用
Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications
浏览论文内容
中文总结 AI 辅助
该研究建立了德西特空间与双曲空间上曲率流、逆曲率流及曲线缩短流解的对应关系，推导了孤子解的判定条件与存在性，明确了非平凡完全孤子的类型。
中文摘要 AI 辅助
我们证明了德西特空间上曲线曲率流的类空解与二维双曲空间上逆曲率流的解一一对应；德西特空间上曲率流的类时解与逆曲率流的类时解一一对应；二维双曲空间上曲线缩短流的解与德西特空间上逆曲率流的类空解一一对应。我们证明，对于德西特空间上的类空曲线，曲率流是弧长泛函的梯度型流。我们发现，德西特空间上的类空或类时曲线是曲率流（对应逆曲率流）的孤子解，当且仅当它的曲率（对应曲率的倒数）可表示为其切向量场与三维闵可夫斯基空间中固定向量v的内积。我们证明，对于每个向量v，德西特空间上的曲率流和逆曲率流各存在一个2参数族的类时（类空）孤子解。我们证明不存在非平凡的完全类时孤子，却存在非平凡的完全类空解。作为曲率流的推论，我们得到了德西特空间和双曲空间上逆曲率流孤子解的行为。
英文摘要
We show that the spacelike solutions to the curvature flow for curves on the De Sitter space are in correspondence with the solutions to the inverse curvature flow on the 2-dimensional hyperbolic space, that on the De Sitter space, the timelike solutions to the curvature flow are in correspondence with the timelike solutions to the inverse curvature flow, and that the solutions curve shortening flow on the 2-dimensional hyperbolic space are in correspondence with the solutions to the spacelike solutions to the inverse curvature flow on the De Sitter space. We prove that, for spacelike curves on the De Sitter space, the curvature flow is a gradient-type flow for the arc-length functional. We observe that a spacelike or timelike curve on the De Sitter space is a soliton solution to the curvature flow (resp. inverse curvature flow) if and only if its curvature (resp. inverse of its curvature) can be written as the inner product between its tangent vector field and a fixed vector $v$ of the 3-dimensional Minkowski space. We prove that for each vector $v$, there exists a 2-parameter family of timelike (spacelike) soliton solutions to the curvature flow and to the inverse curvature flow on the De Sitter space. We show that there exists no non-trivial complete timelike soliton. There exist non-trivial complete spacelike solutions. As a consequence of curvature flow, we obtain the behavior of the soliton solutions to the inverse curvature flow on the De Sitter space and hyperbolic space.
