AI 中文总结
研究针对Z/2调和1-形式的卡拉比手术方法，通过证明相关定理，利用切割粘贴等操作，在弱正则性假设下灵活构造和修改Z/2调和1-形式，并得出连通和、局部替换等定理及相关应用。
AI 中文摘要
我们证明了卡拉比内蕴调和性定理的二值类似物，并利用它引入了卡拉比手术方法，这是一种针对Z/2调和1-形式的手术理论。当允许环境度量变化时，新的Z/2调和形式的构造可归结为切割和粘贴封闭的二值1-形式、匹配局部调和模型以及控制所得奇异叶状结构的可迁性。对于这些构造，奇异胶合中出现的纳什-莫泽型解析变形问题被局部模型匹配和叶状结构上的全局动力学条件所取代。所得过程在弱正则性假设下给出了一种灵活的方式来构造和修改Z/2调和1-形式。作为应用，我们得到了连通和与局部替换定理，通过规定的欧几里得模型爆破孤立的普通零点，分裂光滑的k-非退化分支分量，并使用合适的分辨率模型在三维和四维中使图形奇异集去奇异化。
英文摘要
We prove a 2-valued analogue of Calabi's intrinsic harmonicity theorem and use it to introduce the Calabi surgery method, a surgery theory for $\mathbb{Z}/2$ harmonic $1$-forms. Once the ambient metric is allowed to vary, the construction of new $\mathbb{Z}/2$ harmonic forms can be reduced to cutting and pasting closed 2-valued 1-forms, matching local harmonic models, and controlling the transitivity of the resulting singular foliation. For these constructions, the Nash--Moser-type analytic deformation problem that arises in singular gluing is replaced by local model matching and a global dynamical condition on the foliation. The resulting procedure gives a flexible way to construct and modify $\mathbb{Z}/2$ harmonic 1-forms under weak regularity assumptions. As applications, we obtain connected-sum and local replacement theorems, blow up isolated ordinary zeros by prescribed Euclidean models, split smooth $\vec{k}$-nondegenerate branching components, and desingularize graphic singular sets in dimensions 3 and 4 with suitable resolution models.
Comments45 pages, 3 figures
