This research will focus on function theoretic properties of L-convex manifolds. These L-convex manifolds are complex manifolds which are convex with respect to sections of a bundle L. L-convex manifolds have recently proven useful in attacking some classical problems in homomorphic convexity, such as the Levi problem and Shafarevich's Conjecture on the universal covering space of a projective variety. The project involves five areas of research: exhaustion functions, approximation of sections, embedding, cohomology with values in a coherent analytic sheaf, and characterization by local boundary conditions. This research is in the general area of the geometric analysis of surfaces or manifolds. These manifolds or surfaces exist in two, three, or many dimensions and involve both analytic and geometric techniques to understand and describe their structure. Often the surface is given coordinates and a metric which provides a measurement of distance or angle. In this project the manifolds have coordinates over the complex field and support a hermitian metric.
