本文研究了（1+3）维时空$\mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$中的有限能量等变波映射，其中$\mathbb R \times (\mathbb R \times \mathbb{S}^2)$上的度规为ds ^2 = -dt ^2 + dr ^2+（r ^2 + 1）\left（d θ ^2 + \sin^2 θd φ ^2 \right），\quad t，r \in \mathbb{R}，（θ，φ）\in \mathbb{S}^2 .每个常数时间片都由黎曼流形$\mathcal M := \mathbb R \times \mathbb{S}^2$给出，度量为ds ^2 = dr ^2+（r ^2 + 1）\left（d θ^2 + \sin^2 θd φ^2 \right）。黎曼流形$\mathcal M$在$r \rightarrow \pm \infty$处包含两个渐近欧几里得端点，它们通过面积为$4 π^2$的球形喉道连接。时空$\mathbb R \times \mathcal M$是广义相对论中虫洞几何的一个简单例子。在这项工作中，我们将考虑1-等变或共旋波映射。每个共旋波图可以通过其拓扑度$n$来索引。对于每个$n$，存在唯一的能量最小化共转调和映射$Q_{n} : \mathcal M \rightarrow \mathbb{S}^3$的程度$n$。在这项工作中，我们表明，模的自由辐射项，每一个度$n$的共旋波图强烈收敛到$Q_{n}$。这解决了Bizon和Kahl在共转情况下提出的一个猜想。
英文摘要：
In this paper, we initiate the study of finite energy equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$ where the metric on $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ is given by ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d θ^2 + \sin^2 θd φ^2 \right ), \quad t,r \in \mathbb{R}, (θ,φ) \in \mathbb{S}^2. The constant time slices are each given by the Riemannian manifold $\mathcal M := \mathbb R \times \mathbb{S}^2$ with metric ds^2 = dr^2 + (r^2 + 1) \left ( d θ^2 + \sin^2 θd φ^2 \right ). The Riemannian manifold $\mathcal M$ contains two asymptotically Euclidean ends at $r \rightarrow \pm \infty$ that are connected by a spherical throat of area $4 π^2$ at $r = 0$. The spacetime $\mathbb R \times \mathcal M$ is a simple example of a wormhole geometry in general relativity. In this work we will consider 1--equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree $n$. For each $n$, there exists a unique energy minimizing corotational harmonic map $Q_{n} : \mathcal M \rightarrow \mathbb{S}^3$ of degree $n$. In this work, we show that modulo a free radiation term, every corotational wave map of degree $n$ converges strongly to $Q_{n}$. This resolves a conjecture made by Bizon and Kahl in the corotational case.
Nonexistence of blow-up solutions with smooth radiation for energy-critical equivariant wave maps
Institut de Mathématiques de Jussieu, Sorbonne Université, Université Paris Cité（巴黎数学研究所，索邦大学，巴黎西岱大学）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model
University of Cologne（科隆大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；University of Toronto（多伦多大学）；Yale University（耶鲁大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
