An Elementary Proof of the Existence of Isothermal Parameters on a Surface

Abstract

In order that isothermal parameters exist it is necessary to impose on the metric some regularity assumptions. In fact, it was shown recently by Hartman and Wintnerl that it is not sufficient to assume the functions E, F, G to be continuous. So far the weakest conditions under which the isothermal parameters are known to exist were found by Korn and Lichtenstein.2 To formulate their theorem we recall that a function f(x, y) in a domain D of the (x, y)-plane is said to satisfy a Holder condition of order X, 0 <X _ 1, if the inequality