On Chern minimal surfaces in Hermitian surfaces

In this paper we introduce the Chern minimal surface in Hermitian surfaces by using the Chern connection, and we show that it only has isolated complex and anticomplex points for a generic one (neither holomorphic nor antiholomorphic). For a generic Chern minimal $f$ from compact Riemann surface $\Sigma$ in a Hermitian surface $M$, we establish two identities which related to the sum of the orders of all complex points, anticomplex points denoted by $P$, $Q$ respectively, the cap product of the pull-back of the first Chern class $f^*c_1(M)$ and $[\Sigma]$, the Euler characteristic of tangent bundle $\chi(T\Sigma)$ and the Euler characteristic of normal bundle $\chi(T^\perp\Sigma)$. More precisely, we obtain the formulae $P-Q=-f^*c_1(M)[\Sigma]$ and $P+Q=-\big(\chi(T\Sigma)+\chi(T^\perp\Sigma)\big)$. We also give some applications of these formulae.