The Seiberg–Witten equations on manifolds with boundary II: Lagrangian boundary conditions for a Floer theory

In this paper, we study the Seiberg–Witten equations on the product $\mathbb{R} \times Y$, where $Y$ is a compact $3$-manifold with boundary. Following the approach of Salamon and Wehrheim in Anti-self-dual instantons with Lagrangian boundary conditions I: Elliptic theory [Comm. Math. Phys. 254 (2005), no. 1, 45–89] and Instanton Floer homology with Lagrangian boundary conditions [Geom. Topol. 12 (2008), no. 2, 747–918] in the instanton case, we impose Lagrangian boundary conditions for the Seiberg–Witten equations. The resulting equations we obtain constitute a nonlinear, nonlocal boundary value problem. We establish regularity, compactness, and Fredholm properties for the Seiberg–Witten equations supplied with Lagrangian boundary conditions arising from the monopole spaces studied in The Seiberg-Witten equations on manifolds with boundary I [Comm. Anal. Geom. 20 (2012), no. 3, 565–676]. This work therefore serves as an analytic foundation for the construction of a monopole Floer theory for 3-manifolds with boundary.

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The author was supported by NSF grants DMS-0706967 and DMS-0805841.

Received 30 November 2010

Published 31 January 2018