UNIFORMIZATION THEORY AND 2D GRAVITY .1. LIOUVILLE ACTION AND INTERSECTION-NUMBERS

This is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classica l Liouville action appears in the expression for the correlators of to pological gravity. Next we derive an inequality involving the cutoff o f 2D gravity and the background geometry. Another result, still relate d to uniformization theory, concerns a relation between the higher gen us normal ordering and the Liouville action. We introduce operators co variantized by means of the inverse map of uniformization. These opera tors have interesting properties, including holomorphicity. In particu lar, they are crucial for showing that the chirally split anomaly of C FT is equivalent to the Krichever-Novikov cocycle and vanishes for def ormation of the complex structure induced by the harmonic Beltrami dif ferentials. By means of the inverse map we propose a realization of th e Virasoro algebra on arbitrary Riemann surfaces and find the eigenfun ctions for the holomorphic covariant operators defining higher order c ocycles and anomalies which are related to W algebras. Finally we face the problem of considering the positivity of e(sigma), with a the Lio uville field, by proposing an explicit construction for the Fourier mo des on compact Riemann surfaces. These functions, whose underlying num ber-theoretic structure seems related to Fuchsian groups and to the ei genvalues of the Laplacian, are quite basic and may provide the buildi ng blocks for properly investigating the long-standing uniformization problem posed by Klein, Koebe and Poincare.