Conformal Geometry and Riemann Surfaces

Annotation

The course gives an elementary introduction to basics of the conformal geometry in general dimensions and a more detailed account of Riemann surfaces.  In particular, moduli space of complex structures, their deformations in terms of Beltrami differentials, and the action of Witt algebra are described in some details.

Course plan

1. Conformal maps. Linear conformal transformations of $R^n$. Liouville's theorem on conformal maps in dimension n>2.
2. Inversions as Mobius transformations.  The Mobius group. Generators of the Mobius group.
3. Conformal structure on a manifold (conformal geometry). Conformally equivalent metrics. Symmetries of conformal structure (conformal transformations). Conformal Killing vectors.
4. Flat model of conformal geometry. Realization of conformal transformations as linear isometries of $R(n+1,1)$.
5. Conformal geometry in two dimensions. Local conformal transformations of $R^2$. Holomorphic and antiholomorphic maps. Witt Algebra.
6. Riemann mapping theorem.  Uniqueness of the complex structure on $S^2$.
7. Existence of conformally flat coordinates for an arbitrary metric in 2 dimensions. The Beltrami equation.
8. Complex structures on a torus. The moduli space and the Teichmuller space for a torus.
9. Dimension of the moduli spaces for g>1. Description of tangent space in terms of the Beltrami differentials.
10. The action of the Witt algebra on the moduli space of Riemann surfaces with a marked point and a local parameter therein  (Schiffer-Spencer construction).

Literature

1. R. Nevanlinna, H. Behnke and H. Grauert, L. V. Ahlfors, D. C. Spencer, L. Bers, K. Kodaira, M. Heins, J. A. Jenkins. Analytic Functions. Princeton Mathematical Series No. 24 Hardcover – January 1, 1960 
2. B. A. Dubrovin, A. T. Fomenko, S. P. Novikov. Modern Geometry — Methods and Applications. Part I. Graduate Texts in Mathematics book series.
3. M.Schiffer, Menahem, D. Spencer. Functionals of Finite Riemann Surfaces.
4. Sharpe, R.W. Cartan's Generalization of Klein's Erlangen Program.
5. Eastwood, M. Notes on conformal differential geometry, Suppl. Rendi. Circ. Mat. Palermo, 1996, 43, 57.
6. M. Schottenloher, A Mathematical Introduction to Conformal Field Theory (Lecture Notes in Physics, 759).