2$. Then for every domain $\Omega \subset X$ and every compact set $K \subset \Omega$ with $\Omega \backslash K$ connected, holomorphic or meromorphic functions on $\Omega \backslash
K$ extend holomorphically or meromorphically and uniquely to $\Omega$[:]{} $$\mathcal{O}_X(\Omega\backslash K)
=
\mathcal{O}_X(\Omega)\big\vert_{\Omega\backslash K}
\ \ \ \ \ \ \
\text{\rm or}
\ \ \ \ \ \ \
\mathcal{M}_X(\Omega\backslash K)
=
\mathcal{M}_X(\Omega)\big\vert_{\Omega\backslash K}.$$

Some comments on the hypotheses are in order. Firstly, connectedness of $X$ is not a restriction, since otherwise, $\Omega$ would be contained in a single component of $X$. Secondly, as $X$ is $(n-1)$-complete, $i\, \partial \overline{ \partial}\, \big( \rho \big
\vert_{ X_{ \rm reg}} \big)$ has at least $2$ positive eigenvalues at every point $z\in X_{ \rm reg}$, and consequently, each super-level set $$\big\{z\in X:\, 
\rho(z)>c\big\},$$ has a pseudoconcave boundary at every smooth point $z\in X_{\rm reg}$ with $d\rho ( z) \neq 0$ and in fact, the Levi-form of this boundary has at least one negative eigenvalue at $z$. Thirdly, by a theorem of Ohsawa ([@oh1984]), every (connected) $n$-dimensional [*noncompact*]{} complex manifold is $n$-complete, and in fact, easy examples show that Hartogs extension may fail: take the product $X := R\times S$ of two Riemann surfaces, with $R$ 