AI 中文总结
本文构造了稳态调和映射的反例，其奇异集维数为n-2且零测度，从而否证了余维数至少为三的猜想。
AI 中文摘要
调和映射正则性理论中的一个核心猜想断言，稳态调和映射的奇异集余维数至少为三。本文构造了该民间猜想的反例。我们构造了奇异集 $S$ 满足 $\dim S=n-2$ 且 $\mathcal{H}^{n-2}(S)=0$ 的稳态调和映射。目标流形是一个与 $\mathbb{S}^2 \times \mathbb{T}^3$ 微分同胚的光滑紧致流形，其度量在 $C^\infty$ 拓扑下任意接近标准乘积度量。
英文摘要
A central conjecture in the regularity theory of harmonic maps asserts that the singular set of a stationary harmonic map has codimension at least three. In this paper, we construct counterexamples to this folklore conjecture. We construct stationary harmonic maps whose singular sets $S$ satisfies $\dim S=n-2$ and $\mathcal{H}^{n-2}(S)=0$ . The target is a smooth compact manifold diffeomorphic to $\mathbb{S}^2 \times \mathbb{T}^3$, with metric arbitrarily close in $C^\infty$ to the standard product metric.
