Concentrating local solutions of the two-spinor Seiberg-Witten equations on 3-manifolds
Abstract
Given a compact 3-manifold $Y$ and a $\mathbb{Z}_{2}$-harmonic spinor $(\mathcal{Z}_{0},A_{0},\Phi_{0})$ with singular set $\mathcal{Z}_{0}$, this article constructs a family of local solutions to the two-spinor Seiberg-Witten equations parameterized by $\epsilon\in(0,\epsilon_{0})$ on tubular neighborhoods of $\mathcal{Z}_{0}$. These solutions concentrate in the sense that the $L^{2}$-norm of the curvature near $\mathcal{Z}_{0}$ diverges as $\epsilon\rightarrow0$ and after renormalization they converge locally to the original $\mathbb{Z}_{2}$-harmonic spinor. In a sequel to this article, these model solutions are used in a gluing construction showing that any $\mathbb{Z}_{2}$-harmonic spinor satisfying some mild assumptions arises as the limit of a family of two-spinor Seiberg-Witten solutions on $Y$.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Geometry and complex manifolds
Nonlinear Partial Differential Equations
Publication details
- Journal / source
- Asian Journal of Mathematics
- Publisher
- International Press of Boston
- Volume
- 30
- Issue
- 2
- Pages
- 147–260
- Publication date
- 1 January 2026
- Language
- EN