Singularities and Complexity in CR Geometry

Jiří Lebl
Spring 2007
University of California at San Diego

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 ≥ 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 functions.

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