arXiv 1806.092332018-06math.DG

类光点超曲面

Hypersurfaces with light-like points

Masaaki Umehara, Kotaro Yamada

备注：: 25 pages

摘要：

考虑一个常平均曲率浸入$F:U(\subset \boldsymbol{R}^n)\to M$到任意Lorentz $(n+1)$ -流形$M$中.如果$F$的第一基本形式$ds^2$在$o$处退化，则点$o\in U$称为类光点。我们用$B_F$表示与$ds^2$相关的对称矩阵关于$o$的局部坐标系的行列式函数。如果$B_F$的外导数在$o$处为零，则称类光点$o$为简并点。我们证明了如果$o$是一个退化的类光点，那么$F$的像包含一个$M$的类光测地线段通过$f(o)$（参见\定理E）。这就解释了为什么洛伦兹-闵可夫斯基$(n+1)$ -空间形式$\boldsymbol{R}^{n+1}_1$中的几个已知的常平均曲率超曲面的例子在$F$的适当正则性条件下，在它们的类光点集上包含类光线。并给出了几个相关结果。

英文摘要：

Consider a constant mean curvature immersion $F:U(\subset \boldsymbol{R}^n)\to M$ into an arbitrary Lorentzian $(n+1)$-manifold $M$. A point $o\in U$ is called a light-like point if the first fundamental form $ds^2$ of $F$ degenerates at $o$. We denote by $B_F$ the determinant function of the symmetric matrix associated to $ds^2$ with respect to a local coordinate system at $o$. A light-like point $o$ is said to be degenerate if the exterior derivative of $B_F$ vanishes at $o$. We show that if $o$ is a degenerate light-like point, then the image of $F$ contains a light-like geodesic segment of $M$ passing through $f(o)$ (cf.\ Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz-Minkowski $(n+1)$-space form $\boldsymbol{R}^{n+1}_1$ contain light-like lines on their sets of light-like points, under a suitable regularity condition of $F$. Several related results are also given.

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