| A compact orientable surface M with boundary admits almost-complex structures J on its compressed tangent bundle bTM. We call a surface equipped with such a structure a b-holomorphic complex curve. The interior of a b-holomorphic curve is an ordinary non-compact Riemann surface. But the holomorphic structure at the boundary is singular, in the sense that the curve cannot be realized as an embedded submanifold (with boundary) of a larger Riemann surface.;In this dissertation, we study b-holomorphic structures. We discover invariant or characteristic objects associated to b-holomorphic curves, and others associated to holomorphic line bundles over such curves, including a generalized degree. We then use these invariants to prove classification theorems.;We also investigate the existence of constant-curvature connections on these line bundles. In particular, we provide a necessary and sufficient condition for the existence of a hermitian holomorphic b-connection whose curvature is a constant times the volume form (that is, the volume form induced by a given hermitian metric on the base manifold). Such a connection is an absolute minimum for the Yang-Mills functional on the bundle. |
