Complex structures on Riemann surfaces

Abstract:

Let $X$ be a Riemann surface (compact or noncompact) with the property that the length of every closed geodesic is bounded away from zero. Then we show that sufficiently small complex structures on $X$ can be described without making use of Schwarzian derivatives or the theory of quasiconformal mappings. Instead, we use methods developed in Kuranishi’s work on the existence of locally complete families of deformations of compact complex manifolds. We introduce norms $|\quad {|_k}$ ($k$ a positive integer) on the space of ${C^\infty }(0,p)$-forms with values in the tangent bundle on $X$, which are similar to the usual Sobolev $||\quad |{|_k}$-norms. (In the compact case $|\quad {|_k}$ is equivalent to $||\quad |{|_k}$.) Then we prove that certain properties of $||\quad |{|_k}$, crucial for Kuranishi’s approach, are also satisfied by $|\quad {|_k}$.