Compactness of conformal Chern-minimal surfaces in Hermitian surface

TL;DR

This paper establishes the compactness property of conformal Chern-minimal surfaces in Hermitian surfaces, showing that sequences with bounded area converge to a bubble tree consisting of Chern-minimal maps, preserving area and homotopy class.

Contribution

It proves the bubble tree convergence for sequences of conformal Chern-minimal maps into Hermitian surfaces, extending minimal surface compactness results.

Findings

Sequences with bounded area have bubble tree limits.

Limit maps preserve area and homotopy class.

The limit consists of a main Chern-minimal map and finitely many bubble maps.

Abstract

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.