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Anti-self-dual instantons with Lagrangian boundary conditions II: Bubbling
We study bubbling phenomena of anti-self-dual instantons on \H^2\times\S,
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.Comment: 44 pages. Some corrections and rearrangements in section 5: Theorem
5.1 (now 5.3) was previously stated with incorrect assumption
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