Mild manifolds and a non-standard Riemann existence theorem

Let R be an o-minimal expansion of a real closed field R, and K be the algebraic closure of R. In earlier papers we investigated the notions of R-definable K-holomorphic maps, K-analytic manifolds and their K-analytic subsets. We call such a K-manifold mild if it eliminates quantifers after endowing it with all it K-analytic subsets. Examples are compact complex manifolds and non-singular algebraic curves over K. We examine here basic properties of mild manifolds and prove that when a mild manifold M is strongly minimal and not locally modular then it is biholomorphic to a non-singular algebraic curve over K.