Let $\Sigma$ be a closed orientable surface of genus $g$ with $m$ marked points $x=\{x_1, \ldots, x_m\}$ and $j_0$ denote a complex structure on $\Sigma$. Take a neighborhood $U$ of the isomorphism class of $(\Sigma, x, j_0)$ in the moduli space $\mathcal{M}_{g,m}$ of genus $g$ Riemann surfaces with $m$ marked points.

My question is the following:

Can one take a compact subset $K$ of $\Sigma$ such that $K$ does not contain any marked point, and a representative of each element of a neighbourhood $U$ of $j_0$ can be obtained from $j_0$ by deforming it only on $K$?

Any comment will be helpful. Thank you in advance.