Lars Alexandersson: Research

My monograph doctoral thesis "On Vanishing-Curvature Extensions of Lorentzian Metrics" (chapters 1-3 published in The Journal of Geometric Analysis, 4, 425-466 (1994)) deals with the problem of extending Lorentzian metrics, that are defined on the boundary of a domain in the complex plane, into the interior of the domain in such a way that a certain curvature vanishes. Connections exist with

• the debar equation;
• the Yang-Mills equation;
• set-valued analytic functions;
• interpolation of Banach spaces.

The problem arose in set-valued analytic function theory, developed mainly by Slodkowski. When the values of these functions are circular disks or ellipsoids, it is possible to associate to them a vector bundle with a Lorentzian metric. Properties of the set-valued function are translated into conditions on the metric; e.g., that the graph is polynomially convex corresponds to a flat metric. Surprisingly enough, the very same equation with a two dimensional base domain becomes the Yang-Mills equation! This well-known equation from physics has also been studied by Coifman and Semmes in a situation that resembles mine - their interest stems from the connection with interpolation of Banach spaces.

I have shown that if there exists a certain subsolution (a projectively nonpositive extension), then there also exists a desired solution (a projectively flat one); for one dimensional fibers these concepts correspond to Levi-concave and Levi-flat surfaces, respectively.

In the paper "On Holomorphic Factorization and Meromorphic Selectors" (Bulletin des Sciences Mathématiques, 122, 67-82 (1998)) I showed that when the base domain is the unit disk and the fibers are circular disks, then it is generically possible to find an extension of them (in a natural generalization allowing half planes and the exteriors of circles, i.e., all circular disks in projective space), even if there does not exist any holomorphic section inside the disks on the boundary; in fact, the number of singularities of the extension depends on the number of poles a meromorphic section must have if it is to lie inside the disks on the boundary. The method of proving these results are quite different from the one used in the thesis above; they have more in common with the theory of infinite Hankel matrices and the famous work by Adamyan, Arov and Krein.

A later result, "A Duality Argument for Existence and Uniqueness of Projectively Flat Lorentizan Metrics" dealing with multiply connected domains and ellipsoid fibers is that if there exists a holomorphic section lying inside the ellipsoid fibers on the boundary, then there exists a projectively flat extension of them. The method of proof is based on duality arguments for holomorphic vector bundles.

Currently I am mostly interested in the multi-dimensional Riemann-Hilbert problem, i.e., given a totally real manifold M in n-dimensional complex space, find holomorphic disks (holomorphic mappings of the unit disk) having their boundaries in M. Stay tuned!

If you are interested in my publications, please feel free to contact me!