arXiv 1503.043092015-03math.DG

$\mathbb{S}^4_1$中边缘捕获曲面的共形几何

Conformal geometry of marginally trapped surfaces in $\mathbb{S}^4_1$

Eduardo Hulett

备注：: 23 pages. This a new improved version. Whole sections rewritten with corrected statements and proofs. References added. Comments are welcome

摘要：

类空曲面$S\subset \mathbb{S}^4_1$若其平均曲率向量是类光的，则称其为边缘陷波曲面.在任何定向类空曲面$S \subset \mathbb{S}^4_1$上，我们证明了法丛$ν(S)$的定向选择决定了$G: S \to \mathbb{S}^3$的光滑映射，我们称之为$S$的零高斯映射.我们表明，如果$S$是边际捕获，然后$G$是一个共形浸入远离的零的某些二次霍普夫微分的$S$，所以表面$G(S)$是唯一确定的共形变换的$\mathbb{S}^3$由两个不变量：正常的霍普夫微分$κ$和施瓦茨衍生物$s$。我们发现，这些不变量加上一个额外的二次微分$δ$相关的微分方程，并确定几何的$S$周围的等距$\mathbb{S}^4_1$。这使我们能够得到一个特征的边缘捕获表面$S$的零高斯图像是一个约束的Willmore曲面在$\mathbb{S}^3$在意义上的C.Bohle，G。彼得斯和U.Pinkall [arXiv：math/0411479]。作为这些结果的应用，我们构造并研究了具有非零平行平均曲率向量的边缘陷陷场曲面和具有平坦法丛的边缘陷场曲面的可积非平凡单参数变形。

英文摘要：

A spacelike surface $S\subset \mathbb{S}^4_1$ is marginally trapped if its mean curvature vector is lightlike. On any oriented spacelike surface $S \subset \mathbb{S}^4_1$ we show that a choice of orientation of the normal bundle $ν(S)$ determines a smooth map $G: S \to \mathbb{S}^3$ which we call the null Gauss map of $S$. We show that if $S$ is marginally trapped then $G$ is a conformal immersion away the zeros of certain quadratic Hopf-differential of $S$ and so the surface $G(S)$ is uniquely determined up to conformal transformations of $\mathbb{S}^3$ by two invariants: the normal Hopf differential $κ$ and the Schwartzian derivative $s$. We show that these invariants plus an additional quadratic differential $δ$ are related by a differential equation and determine the geometry of $S$ up to ambient isometries of $\mathbb{S}^4_1$. This allows us to obtain a characterization of marginally trapped surfaces $S$ whose null Gauss image is a constrained Willmore surface in $\mathbb{S}^3$ in the sense of C.Bohle, G. Peters and U.Pinkall [arXiv:math/0411479]. As an application of these results we construct and study integrable non-trivial one-parameter deformations of marginally trapped surfaces with non-zero parallel mean curvature vector and those with flat normal bundle.

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