Anti-self-dual instantons with Lagrangian boundary conditions II: Bubbling
arXiv:math/0401377 · doi:10.1007/s00220-005-1349-y
Abstract
We study bubbling phenomena of anti-self-dual instantons on $\H^2\times§$, where is a closed Riemann surface. The restriction of the instanton to each boundary slice , $z\in\pd\H^2$ is required to lie in a Lagrangian submanifold of the moduli space of flat connections over that arises from the restrictions to the boundary of flat connections on a handle body. We establish an energy quantization result for sequences of instantons with bounded energy near : Either their curvature is in fact uniformly bounded in a neighbourhood of that slice (leading to a compactness result) or there is a concentration of some minimum quantum of energy. We moreover obtain a removable singularity result for instantons with finite energy in a punctured neighbourhood of . This completes the analytic foundations for the construction of an instanton Floer homology for 3-manifolds with boundary. This Floer homology is an intermediate object in the program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology-3-spheres. In the interior case, for anti-self-instantons on , our methods provide a new approach to the removable singularity theorem by Sibner-Sibner for codimension 2 singularities with a holonomy condition.
44 pages. Some corrections and rearrangements in section 5: Theorem 5.1 (now 5.3) was previously stated with incorrect assumptions