On holomorphic matrices on bordered Riemann surfaces

TL;DR

This paper proves that any holomorphic matrix-valued function on a bordered Riemann surface can be factorized into exponentials of holomorphic traceless matrices, extending previous results from the unit disk to more general surfaces.

Contribution

It extends the factorization result of holomorphic matrices from the unit disk to arbitrary compact bordered Riemann surfaces.

Findings

Holomorphic matrix functions on bordered Riemann surfaces can be factorized into exponentials of holomorphic traceless matrices.

The extension from the unit disk to general bordered Riemann surfaces is established.

Supports the broader applicability of matrix factorization techniques in complex analysis.

Abstract

Let $\D$ be the unit disk. Kutzschebauch and Studer \cite{KS} recently proved that, for each continuous map $A:\overline D\to \mathrm{SL}(2,\C)$, which is holomorphic in $\D$, there exist continuous maps $E,F:\overline \D\to \mathfrak{sl}(2,\C)$, which are holomorphic in $\D$, such that $A=e^Ee^F$. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.