Minimal Surface Linear Combination Theorem

Given two univalent harmonic mappings f1 and f2 on D, which lift to minimal surfaces via the Weierstrass-Enneper representation theorem, we give necessary and sufficient conditions for f3 = (1−s)f1+sf2 to lift to a minimal surface for s ∈ [0, 1]. We then construct such mappings from Enneper’s surface to Scherk’s singularly periodic surface, Sckerk’s doubly periodic surface to the catenoid, and the 4-Enneper surface to the 4-noid. 1 Background Complex-valued harmonic mappings can be regarded as generalizations of analytic functions. In particular, a harmonic mapping is a complexvalued function f = u+ iv, where the C2 functions u and v satisfy Laplace’s equation. The Jacobian of such a function is given by Jf = uxvy−uyvx. On a simply connected domain D ⊂ C, a harmonic mapping f has a canonical decomposition f = h+ g, where h and g are analytic in D, unique up to a constant [2]. We will only consider harmonic mappings that are univalent with positive Jacobian on D = {z : |z| < 1}. The dilatation ω of a harmonic map f is defined by ω ≡ g′/h′. A result by Lewy [10] states that |h′(z)| > |g′(z)| if and only if f = h+ ḡ is sense–preserving and locally univalent. The reader is referred to [6] for many interesting results on harmonic mappings. One area of study is the construction of families of harmonic mappings [7] and their corresponding minimal surfaces [1], [4], [5]. We now present some necessary background concerning minimal surfaces. Let M be an orientable surface that arises from a differentiable mapping x from a domain V ⊂ R2 (or C) into R3, so that x(u, v) = (x1(u, v), x2(u, v), x3(u, v)). The parametrization x is isothermal (or conformal) if and only if xu · xv = 0 and xu · xu = xv · xv(= λ > 0). Note that there exists an isothermal parametrization on any regular minimal surface (see [3]). Fix a point p on M . Let t denote a 1 vector tangent to M at p and n the unit normal vector to M at p. Then t and n determine a plane that intersects M in a curve γ. The normal curvature κt at p is defined to have the same magnitude as the curvature of γ at p with the sign of κt chosen to be consistent with the choice of orientation of M . The principal curvatures, κ1 and κ2, of M at p are the maximum and minimum of the normal curvatures κt as t ranges over all directions in the tangent space. The mean curvature of M at p is the average value H = 12 (κ1 + κ2). Definiton 1. A minimal surface in R3 is a regular surface for which the mean curvature is zero at every point. The following standard theorem provides the link between harmonic univalent mappings and minimal surfaces: Theorem 2. (Weierstrass-Enneper Representation). Every regular minimal surface has locally an isothermal parametric representation of the form