Studying the solution set of most geometric differential equations on a non-closed manifold X, with a boundary or a cylindrical end, is often challenging. Contributions from the boundary of X often arise and demand special attention. In this talk, I will explain some of these problems in the example of Seiberg-Witten equations on an open manifold X which is the complement of a Riemann surface in a closed manifold Y. In particular, going from the closed manifold Y to the open subset X, we lose ellipticity of equations, we face problems with the convergence of solutions (compactness), and several other issues. I will explain the classical approach to working around these problems and discuss novel ideas using logarithmic structures in differential geometry.
This is a project in progress with P. Safari.