Harmonic functions on Riemann surfaces

Volodymyr Sharko
Contents of lectures



1) Riemann surfaces and their topological classification.

2) Conformal structures, the theorem on uniformisation, description of the
set of conformal structures on a Riemann  surface of genus g.

3) The definition of Laplas-Beltrami operator on Riemann surfaces and
its properties.

4) Harmonic functions on a Riemann  surface:
   a) theorems of existence
   b) local in the neighborhood of critical isolated point
   c) Morse inequalities for critical points

5) Construction of level set of harmonic function

6) Problems of existence of harmonic functions with predefined properties

7) Open questions and conjectures