具有锥形奇点的调和映射的形变

中文摘要 AI 辅助

我们研究了紧致黎曼流形之间具有孤立奇点的调和映射的形变理论，前提是所有切映射（在原点之外）都是光滑的，并且向切映射的衰减是多项式的。我们将这类映射称为锥形奇点调和映射。在关于切映射的 Morse 指标和零化度以及 Jacobi 算子的加权核的非退化假设的某些条件下，我们证明了这类调和映射在背景度量的微小 $C^2$ 扰动下仍然存在。在论文的第二部分，我们构造了满足我们形变定理假设的锥形奇点调和映射的例子。我们首先证明了径向投影 $\mathbb{R}^m \setminus \{0\} \to \mathbb{S}^{m-1}$（$m \geq 4$）和复 Hopf 纤维化 $\mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^{n}$（$n \geq 1$）满足关于切映射的期望条件。然后我们构造了锥形奇点映射 $\mathbb{S}^{m} \setminus \{N,S\} \to \mathbb{S}^{m-1}$、$\mathbb{S}^{4} \setminus \{N,S\} \to \mathbb{S}^2$ 和 $\mathbb{CP}^2 \setminus \{[0:0:1]\} \to \mathbb{CP}^1$ 的显式例子，这些映射是调和的，并且相对于定义域上的适当度量和目标上的圆度量满足形变定理的假设。

英文摘要

We study the deformation theory of harmonic maps with isolated singularities between compact Riemannian manifolds, in the case where all the tangent maps are smooth (away from the origin) and the decay to the tangents is polynomial. We will refer to these as conically singular harmonic maps. Under certain conditions on the Morse index and the nullity of the tangent maps and a non-degeneracy assumption on a weighted kernel of the Jacobi operator, we prove that such harmonic maps persist under a small $C^2$-perturbation of the background metric. In the second part of the paper, we construct examples of conically singular harmonic maps satisfying the assumptions of our deformation theorem. We first prove that the desired conditions on the tangent maps are satisfied by the radial projections $\mathbb{R}^m \setminus \{0\} \to \mathbb{S}^{m-1}$ ($m \geq 4$) and the complex Hopf fibrations $\mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^{n}$ ($n \geq 1$). We then construct explicit examples of conically singular maps $\mathbb{S}^{m} \setminus \{N,S\} \to \mathbb{S}^{m-1}$, $\mathbb{S}^{4} \setminus \{N,S\} \to \mathbb{S}^2$ and $\mathbb{CP}^2 \setminus \{[0:0:1]\} \to \mathbb{CP}^1$ which are harmonic and satisfy the assumptions of the deformation theorem with respect to an appropriate metric on the domain and the round metric on the target.

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78 pages, 1 appendix. Comments welcome!