arXiv 1902.089522019-02math.DGgr-qcmath.AP

（1+2）Minkowski空间中类时极大曲面的嵌入性

Embeddedness of timelike maximal surfaces in (1+2) Minkowski space

E Adam Paxton

摘要：

证明了如果$ϕ\colon \mathbb{R}^2 \to \mathbb{R}^{1+2}$是中曲率为零的光滑真类时浸入，则$ϕ$必然是嵌入，并且$ϕ(\mathbb{R}^2)$的每个紧子集都是光滑图.由此可见，如果演化任何光滑的自相交类空曲线（或任何平面类空曲线，其单位切向量跨越一个封闭的半圆），以便在$\mathbb{R}^{1+2}$中追踪一个中曲率为零的类时曲面，那么演化曲面要么不能保持类时，要么不能保持光滑。我们表明，即使允许零点，这样的柯西演化必须经历一个数量曲率爆破-其中爆破是相对于一个$L^1L^\infty$规范-因此，不断发展的表面将是$C^2$不可扩展的奇异时间。此外，我们还研究了等温规范中自相交曲线演化的单位切线的连续性，它定义了一个著名的超越奇异时间的演化。

英文摘要：

We prove that if $ϕ\colon \mathbb{R}^2 \to \mathbb{R}^{1+2}$ is a smooth proper timelike immersion with vanishing mean curvature, then necessarily $ϕ$ is an embedding, and every compact subset of $ϕ(\mathbb{R}^2)$ is a smooth graph. It follows that if one evolves any smooth self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed semi-circle) so as to trace a timelike surface of vanishing mean curvature in $\mathbb{R}^{1+2}$, then the evolving surface will either fail to remain timelike, or it will fail to remain smooth. We show that, even allowing for null points, such a Cauchy evolution must undergo a scalar curvature blow-up---where the blow-up is with respect to an $L^1L^\infty$ norm---and thus the evolving surface will be $C^2$ inextendible beyond singular time. In addition we study the continuity of the unit tangent for the evolution of a self-intersecting curve in isothermal gauge, which defines a well-known evolution beyond singular time.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文