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This is the first part of an investigation concerning the formulation of 2Dgravity in the framework of the uniformization theory of Riemann surfaces. As afirst step in this direction we show that the classical Liouville actionappears in the expression of the correlators of topological gravity. Next wederive an inequality involving the cutoff of 2D gravity and the backgroundgeometry. Another result, always related to uniformization theory, concerns arelation between the higher genus normal ordering and the Liouville action.Furthermore, we show that the chirally split anomaly of CFT is equivalent tothe Krichever-Novikov cocycle. By means of the inverse map of uniformization wegive a realization of the Virasoro algebra on arbitrary Riemann surfaces andfind the eigenfunctions for {\it holomorphic} covariant operators defininghigher order cocycles and anomalies which are related to $W$-algebras. Finallywe attack the problem of considering the positivity of $e^\sigma$, with$\sigma$ the Liouville field, by proposing an explicit construction for theFourier modes on compact Riemann surfaces.Comment: 53 pages. '95 publ. version, contains Eq.(5.23), independently  derived in hep-th/0004194 studying the null compactification of  type-IIA-strin

UNIFORMIZATION THEORY AND 2D GRAVITY I: LIOUVILLE ACTION AND INTERSECTION NUMBERS