Let $U$ a bordered Riemann surface of genus $g$ with $n-1$ punctures and one hole (i.e., the border has one connected component). Is the following statement true:

"For any punctured Riemann surface $\Sigma$ of of genus $g$ with $n$ punctures there exists an embedding (i.e., an injective holomorphic map) $\Phi: U \rightarrow \Sigma$."?

The motivation for the question is that the statemant would be equivalent to saying that, if one fixes a parameterization of the boundary of $U$, then for every Riemann surface $\Sigma$ of type $(g,n)$ there exists a punctured disk $D$ with parameterized boundary such that $D#U$, which denotes the Riemann surface obtained by gluing $D$ and $U$ along their boundary (cf. http://arxiv.org/abs/1001.5211), is conformally equivalent to $\Sigma$.

Evidence that the statement may be true is coming from the Virasoro uniformization (cf. http://arxiv.org/abs/1312.1562) which appears to be similar to an infinitesimal version of the statement (but uses variations of the complex structure on a fixed surface).


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