On Chern minimal surfaces in Hermitian surfaces

TL;DR

This paper introduces Chern minimal surfaces in Hermitian surfaces, establishing identities relating complex points, Chern classes, and Euler characteristics, with applications to geometric analysis.

Contribution

It defines Chern minimal surfaces using the Chern connection and derives new identities linking complex points, Chern classes, and Euler characteristics in this context.

Findings

Derived formulas relating complex points and Chern classes.

Established identities connecting Euler characteristics and complex points.

Provided applications of the main identities in geometric analysis.

Abstract

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.