Vol. 21, No. 1, 1967

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Surfaces harmonically immersed in E3

Tilla Weinstein

Vol. 21 (1967), No. 1, 79–87

In this paper we study surfaces in E8 which satisfy conditions necessary and/or sufficient to insure their harmonic immersion with respect to a fixed but not necessarily ordinary conformal structure. Our consideration of such surfaces is based upon the notion that surfaces which share some essential property of minimal surfaces are bound to be interesting. Thus our use here of nonstandard conformal structures is simply a device for the identification of such a class of surfaces distinct from others already much studied, such as quasiminimal surfaces or surfaces of constant mean curvature. In the end, any such endeavor serves to distinguish those facts about minimal surfaces which are special to them from among the many facts which apply to larger classes of surfaces sharing some one vital property of minimal surfaces.

The more quotable results in this paper refer to a conformal structure RΛ determined by a fixed positive definite linear combination Λ = fI + gΠ of the fundamental forms on the surface, with f and g smooth functions. Specifically, we show that mean curvature H cannot be bounded away from zero on a complete RΛ-harmonically immersed surface in E3. This result is less general than it might seem. For we also prove that where H0 on an RΛ-harmonically immersed surface, ΛαII, with Πdefined by √H2--−-K-II= HII KI. Included is an example of an RΛ-harmonically immersed surface on which H0.

Mathematical Subject Classification
Primary: 53.75
Received: 14 December 1965
Published: 1 April 1967
Tilla Weinstein