An elementary proof of the existence of isothermal parameters on a surface

Abstract

shiing-shen chern 1. Introduction.Let (1) ds2 = E(x, y)dx2 + 2F(x, y)dxdy + G(x, y)dy2, EG -F2 > 0, E > 0, be a positive definite Riemann metric of two dimensions defined in a neighborhood of a surface with the local coordinates x, y.By isothermal parameters we mean local coordinates u, v relative to which the metric takes the formIn 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 Wintner1 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.2To formulate their theorem we recall that a function/(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(3) | f(x, y) -f(x', y') | < CV holds for any two points (x, y), (x', y') of D, where C is a constant and r is the Euclidean distance between these two points.With this definition the theorem of Korn-Lichtenstein can be stated as follows:

Locations

• Proceedings of the American Mathematical Society - View - PDF