An elementary proof of the existence of isothermal parameters on a surface
Shiing-Shen Chern · Proceedings of the American Mathematical Society · 1955
Cf. also the paper of C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans.Amer.Math.Soc.vol.43 (1938) pp.126-166, in which the isothermal parameters are shown to exist in a generalized sense under weaker hypotheses.Added in proof.Weaker conditions were recently found by Hartman and Wintner.Cf. their paper, On uniform Dini conditions in the theory of linear partial differential equations of elliptic type, Amer.