Abstract.We study the Liouville equation Au = -e2u in the complex plane with prescribed singularities and obtain a necessary and sufficient condition for the existence of the solution. The proof is based on the continuity method and a uniqueness theorem.Suppose M is a punctured Riemann sphere with « points Vx, ... ,Vn removed and ax, ... , a" £ (0, 2n) are « given numbers. We are interested in seeking a convex polytope with boundary P in S3 having n vertices so that P -{V\, ... ,Vn} is the Riemann surface M and the cone angle at V¡ is a,. Our result is the following Theorem 1. Let M be an n-punctured Riemann sphere (« > 3) and a, be « numbers in (0, 2n). Then there is a (necessarily unique) convex polytope in S3 with « vertices whose boundary P satisfies (a) P -{ Vx, ... , Vn} is conformally equivalent to M and (b) the cone angle at V¡ is a¡, if and only if n (1) ¿a,>27r(«-2), ,= i and (2) Y^ai~aj < 2n(n-2), for all j = 1,2, ... , «.Since the cone angle of \z\al2n~x\dz\ at 0 is a, in terms of the singular metric, the theorem can be stated as Theorem 2. The Liouville equation Au = -exp(2u) in the punctured complex plane M -C -{Vx, ... , V"} so that near each V¡, u(z) = /?,log\z -V¡\ + a continuous function, where /?, £ (-1,0) and u = -21og|z| + a continuous