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 H≠0 on an RΛ-harmonically
immersed surface, ΛαII′, with Π′ defined by II′ = HII − KI.
Included is an example of an RΛ-harmonically immersed surface on which
H≠0.
