# Embed a bordered Riemann surface into punctured Riemann surfaces?

Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ punctures, does there necessarily exist an embedding (i.e. an injective holomorphic map) $\Phi: U \to \Sigma$?

The answer is no. For $g=0$, the problem is equivalent to the following: is there a univalent function in the unit disk which takes given values at finitely many given points. It is well known that the answer is no: there are many inequalities relating the points and the values. For example, Theorem 1 on p. 119 of Goluzin's book Geometric theory of functions of a complex variable, Engl. tranls. AMS, 1969 (available on the Internet). Certainly, similar situation prevails for other genera.