Research field: Several Complex Variables and Complex Geometry
Professor: Kang-Hyurk Lee
Rearch description:
My research theme is the uniformization of negatively curved compact Kähler manifolds which is to study the multi-dimensional analog for the geometry of the unit disc its compact quotient spaces.
The uniformization theorem of Riemann surfaces is the classification of simply connected Riemann surfaces: a simply connected Riemann surface is conformally equivalent to one of three model surfaces: the unit disc, the complex plane and the Riemann sphere. This implies that any Riemann surface should be a quotient of a model surface; thus it provides a fundamental methodology for the classification of Riemann surfaces. By the Schwarz-Pick lemma and the Gauss-Bonnet theorem, most Riemann surfaces (especially compact Riemann surface with genus greater than 1) is covered by the unit disc.
In general dimension, the generalization of the uniformization corresponding the unit disc is to understand universal coverings of compact Kähler manifolds whose first Chern class is negative. For the theory of such uniformization, I have been studying following topics
1. The characterization of bounded symmetric domain as a covering of compact Kähler manifolds
2. The potential rescaling method and the existence of a complete holomorphic vector field
3. The action of the automorphism groups of domains
4. Unbounded representation of symmetric and homogeneous domains