Flat fronts in hyperbolic 3-space and their caustics

After Galvez, Martinez and Milan discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space, the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H^3. In the present work we broaden the notion of completeness to weak completeness, and of front to p-front. As a front is a p-front and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces. This more general class contains the caustics of flat fronts -- shown also to be flat by Roitman (who gave a holomorphic representation formula for them) -- which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only p-fronts. Using the new framework, we obtain characterizations for caustics.

在加尔韦斯（Galvez）、马丁内斯（Martinez）和米兰（Milan）发现了双曲3维空间中平坦曲面的（魏尔斯特拉斯型）全纯表示公式之后，本文的第一、第三和第四作者给出了在\(H^3\)中具有奇点的完备平坦前沿的一个框架。在当前的工作中，我们将完备性的概念拓宽到弱完备性，并将前沿的概念拓宽到\(p\)-前沿。由于前沿是\(p\)-前沿且完备性意味着弱完备性，这里的新框架和结果适用于更一般的平坦曲面类。这个更一般的类包含平坦前沿的焦散——罗伊特曼（Roitman）也证明了它们是平坦的（他给出了它们的一个全纯表示公式）——这是一类重要的曲面，通常不是完备的，而只是弱完备的。此外，尽管平坦前沿具有全局定义的法线，但焦散可能没有，这使得它们只是局部的平坦前沿，因此只是\(p\)-前沿。利用新框架，我们得到了焦散的特征描述。