AI 中文总结
本文通过构造反例否定 Ehlers--Kundt 猜想，证明引力 pp-波中测地完备性是普遍的，并引入万有 pp-波概念，证明存在完备的万有 pp-波，同时给出多项式情形的新证明。
AI 中文摘要
本文关注引力 pp-波时空的几何性质。我们给出了 Ehlers--Kundt 猜想的一个反例。事实上，我们证明在 Baire 范畴意义下，测地完备性在引力 pp-波中是普遍的。我们还定义了“万有 pp-波”的概念，即其度量在任意大的紧集上任意接近其他所有引力 pp-波的度量，并证明了存在测地完备的万有 pp-波。我们的方法还足以给出 Ehlers--Kundt 猜想多项式情形的一个新证明（该情形此前由 Flores 和 Sánchez 证明）。
英文摘要
This paper concerns the geometry of gravitational pp-wave spacetimes. We give a counterexample to the Ehlers--Kundt conjecture. In fact, we show that geodesic completeness is generic, in the sense of Baire category, among gravitational pp-waves. We also define a concept of ``universal pp-wave'', i.e. a pp-wave whose metric approximates that of every other gravitational pp-wave arbitrarily closely on arbitrarily large compact sets, and we show that there are geodesically complete universal pp-waves. Our machinery also suffices to give a new proof of the polynomial case of the Ehlers--Kundt conjecture (previously proved by Flores and Sánchez).
