We study the extension problem of holomorphic maps $\sigma :H\to X$ of a Hartogs domain $H$ with values in a complex manifold $X$. For compact Kähler manifolds as well as various non-Kähler manifolds, the maximal domain ${\Omega }_{\sigma }$ of extension for $\sigma$ over $\Delta$ is contained in a subdomain of $\Delta$. For such manifolds, we define, in this paper, an invariant Hex${}_n\left(X\right)$ using the Hausdorff dimensions of the singular sets of $\sigma$’s and study its properties to deduce informations on the complex structure of $X$.