## 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.

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!**

Sidansvarig: lars.alexandersson@liu.se

Senast uppdaterad: 2011-08-23