Harmonically Immersed Surfaces of R n

Abstract

Some generalizations of classical results in the theory of minimal surfaces f: M Rn are shown to hold in the more general case of harmonically immersed surfaces. Introduction. Let (M, g) be a connected Riemann surface with a prescribed metric g in its conformal class and let f: M > Rn be an immersion. It is well known that f realizes M, with the induced metric from Rn, as a minimal surface if and only if f is a conformal (with respect to g) harmonic map (cf., for example, [3 or 8]). That is, the theory of minimal surfaces is substantially the theory of conformally immersed harmonic surfaces. Our purpose is to analyze the case when f is simply a harmonic immersion, to introduce an appropriate Gauss map and, as the main achievement, to establish in this new setting the analogue of three fundamental results (cf. Theorems 1.1 and 2.1, and §3) in the theory of minimal surfaces: the harmonicity of the Gauss map (Ruh and Vilms [13]), equidistribution properties of the Gauss map in cPn (Chern and Osserman [1 and 11]), and the Enneper-Weierstrass representation formulas recently due, for arbitrary codimension, to Hoffman and Osserman [7]. The last two topics (the third for n = 3) have alrealy been treated by T. K. Milnor (see for instance her survey article [9]), but as we remark in §2, our results complement hers in an interesting way. The method of the moving frame as well as the Einstein summation convention are used throughout this paper. 1. The Gauss map and first properties. Let (M, g) be as in the Introduction. We fix the index ranges 1 Rn be an immersion. A Darboux frame al g f is a map E: U c M E(n), U ope in M, such that Received by the editors April 6, 1987. Selected results of this paper were presented by the first named author November 1, 1987 at the 837th meeting of the American Mathematical Society held at Lincoln, Nebraska. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A07, 53A10.