On holomorphic matrices on bordered Riemann surfaces

Abstract

Let 𝔻 be the unit disk. Kutzschebauch and Studer (Bull. Lond. Math. Soc. 51 (2019) 995–1004) recently proved that, for each continuous map 𝐴:𝔻⎯⎯⎯→SL(2,ℂ), which is holomorphic in 𝔻, there exist continuous maps 𝐸,𝐹:𝔻⎯⎯⎯→𝔰𝔩(2,ℂ), which are holomorphic in 𝔻, such that 𝐴=𝑒𝐸𝑒𝐹. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.