The Seiberg-Witten Equations on Manifolds with Boundary II: Lagrangian Boundary Conditions for a Floer Theory

TL;DR

This paper develops an analytic framework for Seiberg-Witten equations on 3-manifolds with boundary, imposing Lagrangian boundary conditions to enable the construction of a monopole Floer theory in this setting.

Contribution

It introduces Lagrangian boundary conditions for Seiberg-Witten equations on manifolds with boundary and establishes their regularity, compactness, and Fredholm properties.

Findings

Proved regularity of solutions with Lagrangian boundary conditions

Established compactness results for the boundary value problem

Demonstrated Fredholm properties essential for Floer theory construction

Abstract

In this paper, we study the Seiberg-Witten equations on the product R x Y, where Y is a compact 3-manifold with boundary. Following the approach of Salamon and Wehrheim 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 [20]. This work therefore serves as an analytic foundation for the construction of a monopole Floer theory for 3-manifolds with boundary.