Singularities and Complexity in CR Geometry

Several related questions in CR geometry are studied. First, the structure of the singular set of Levi-flat hypersurfaces is investigated. The singularity is completely characterized when it is a submanifold of codimension 1, and partial information is gained about higher codimension cases. Second, a local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker property, not being contained inside a possibly singular real-analytic Levi-flat hypersurface is studied and characterized, and a sufficient and necessary condition is given in terms of normal coordinates. One natural generalization of this problem is the classification of codimension 2 real- analytic CR submanifolds, which are locally the boundaries of smooth Levi-flat hypersurfaces. These submanifolds are completely classified in terms of their normal coordinate representation. In fact, an extension theorem is proved allowing smooth Levi-flat hypersurfaces to always be extended past CR submanifolds and in most cases forcing such hypersurfaces to be real-analytic. Examples are found that this extension result is optimal. Finally, relation of the complexity of a mapping and the source and target dimensions is studied for proper holomorphic mappings between balls in different dimensions. A conjecture of John D'Angelo states that a mapping from n to N dimensions has degree less than or equal to (N-1)/(n-1), as long as n is greater than or equal to 3, and 2N-3 when n = 2. The special, but highly nontrivial, case of monomial mappings and a related problem in real algebraic geometry is studied and a weaker bound is proved. The more general cases of polynomial and rational mappings is also treated. In the general rational case, this problem can be thought of as a generalization of the local uniqueness property studied before to vector valued holomorphic function