Compactness of conformal Chern-minimal surfaces in Hermitian surface

The Chern-minimal surfaces in Hermitian surface play a similar role as minimal surfaces in K\"ahler surface (see \cite{[PX-21]}) from the viewpoint of submanifolds. This paper studies the compactness of Chern-minimal surfaces. We prove that any sequence $\{f_n\}$ of conformal Chern-minimal maps from closed Riemann surface $(\Sigma,\emph{\texttt{j}})$ into a compact Hermitian surface $(M, J, h)$ with bounded area has a bubble tree limit, which consisting of a Chern-minimal map $f_0$ from $\Sigma$ into $M$ and a finite set of Chern-minimal maps from $S^2$ into $M$. We also show that the limit preserves area and homotopy class.