Transforms for Non-conformal Harmonic Surfaces in $$\varvec{R^3}$$ R 3

A representation formula for non-conformal harmonic surfaces in $R^3$

“…Remark 1.1. In fact, in [11], we showed that f ± are holomorphic or antiholomorphic. But in this paper, we will see that they are simultaneously holomorphic.…”

“…f − ) has to be constant. A Liouville type theorem was stated in [11]. Examples 5.3 and 5.4 give nonconformal harmonic surfaces f : C → R 3 such that both f + and f − omit just two points on S 2 .…”

“…In [11], the second named author of the present paper gives (±)transforms from a non-conformal harmonic surface f in R 3 , to holomorphic maps f ± into S 2 . It is as an extension of the classical fact that the Gauss map of a minimal surface in R 3 is holomorphic.…”

“…As a national generalization of minimal surfaces, harmonic surfaces have been studied (see [1], [4], [5], [6], [7], [11] and references therein). In this paper, we study non-conformal harmonic surfaces in the Euclidean 3-space R 3 , from the view point of transforms and representation.…”

“…Let us consider the converse problem to [11]. Namely, for two distinct holomorphic maps g, h : M → S 2 where M is a Riemann surface, does there exist a non-conformal harmonic surface f : M → R 3 such that f + = g and f − = h?…”