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Log-Riemann Surfaces

We introduce the notion of log-Riemann surfaces. These are Riemann surfaces given by cutting and pasting planes together isometrically, and come equipped with a holomorphic local diffeomorphism to C called the projection map, and a corresponding flat metric obtained by pulling back the Euclidean metric. We define ramification points to be the points added in the metric completion of the surface with respect to the induced path metric; any such point has a well-defined order $1 \leq n \leq +\infty$ such that the projection map restricted to a small punctured neighbourhood of the point is an $n-to-1$ covering of a punctured disk in C. We prove that simply connected log-Riemann surfaces with finitely many ramification points are biholomorphic to C and the uniformization, with respect to the distinguished charts on the surface given by the projection map, is given by an entire function of the form $F(z) = \int Q(z)e^{P(z)} dz$ where $P, Q$ are polynomials of degrees equal to the number of infinite and finite order ramification points respectively. We also develop an algebraic theory for such log-Riemann surfaces, defining a ring of functions on the surface with finite values at all ramification points, such that the ring separates all points including the infinite order ramification points.