R^n 上 Z/2-调和 1-形式的预定奇异集

中文摘要 AI 辅助

Z/2 调和 1-形式在规范理论和校准几何中自然地作为奇异极限出现，但其奇异集中可能出现的拓扑结构却鲜为人知。本文研究当环境黎曼度量允许变化时这些奇异集的灵活性。我们证明，在维数至少为三的欧几里得空间中，每个具有平凡法丛的紧致光滑余维二子流形都可以实现为非退化 Z/2 调和 1-形式的奇异集，该形式对应的完备度量在紧集外是欧几里得的。作为应用，我们利用 Calabi 手术在具有正第一贝蒂数的闭流形上构造具有预定局部奇异集的 Z/2-调和 1-形式。

英文摘要

Z/2 harmonic 1-forms arise naturally as singular limits in gauge theory and calibrated geometry, but the topology that can occur in their singular sets is poorly understood. In this paper we study the flexibility of these singular sets when the ambient Riemannian metric is allowed to vary. We prove that every compact smoothly embedded codimension-two submanifold with trivial normal bundle in a Euclidean space of dimension at least three can be realized as the singular set of a nondegenerate Z/2 harmonic 1-form for a complete metric that is Euclidean outside a compact set. As an application, we use Calabi surgery to construct Z/2-harmonic 1-forms with prescribed local singular sets on closed manifolds of positive first Betti number.

发表机构

• AMSS（中国科学院数学与系统科学研究院）
• Oxford University（牛津大学）

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