Flat surfaces with singularities in Euclidean 3-space
Flat surfaces with singularities in Euclidean 3-space
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DOI：
10.4310/jdg/1246888486
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发表时间：
2006-05
影响因子：
2.5
通讯作者：
Satoko Murata;M. Umehara
中科院分区：
文献类型：
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作者：
Satoko Murata;M. Umehara
欧氏空间中的完备平面是空间曲线上的柱面，这是经典的知识。这意味着研究平面的整体行为也需要研究奇点。如果一个平坦曲面$f$允许奇异性，但它的高斯映射$\nu$可以光滑地延伸到奇异集上，则$f$被称为波前。另外，如果对$（f，\nu）$给出浸入，则$f$被称为前沿。如果高斯映射处处退化，则称前f$是平坦的。平面波前$f$的平行面和焦面也是平面波前。本文将经典的完备性概念推广到平面字体，并给出了完备平面字体的一个表示公式。作为一个应用，我们证明了一个完整的平坦的前面有适当的嵌入结束当且仅当它的高斯图像是凸曲线。此外，我们还证明了在这种具有嵌入端的平坦正面上存在至少四个除尖边之外的奇异点，这是经典的凸平面曲线四顶点定理的变体。
It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $\nu$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,\nu)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fonts, and give a representation formula for complete flat fronts. As an application, we show that a complete flat front has properly embedded ends if and only if its Gauss image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves.