Harmonic Univalent Mappings and Linearly Connected Domains

We investigate the relationship between the univalence of f and of h in the decomposition f = h+g of a sense-preserving harmonic mapping defined in the unit disk D ⊂ C. Among other results, we determine the holomorphic univalent maps h for which there exists c > 0 such that every harmonic mapping of the form f = h + g with |g| < c|h| is univalent. The notion of a linearly connected domain appears in our study in a relevant way. A planar harmonic mapping is a complex-valued harmonic function f(z), z = x + iy, defined on some domain Ω ⊂ C. When Ω is simply connected, the mapping has a canonical decomposition f = h + g, where h and g are analytic in Ω. Since the Jacobian of f is given by |h| − |g|, it is locally univalent and orientation-preserving if and only |g| < |h|, or equivalently if h(z) 6= 0 and the dilatation ω = g/h has the property |ω(z)| < 1 in Ω. Fundamental questions regarding univalent harmonic mappings are still to be resolved, including important coefficient estimates, and the exact nature of the analogue of the Riemann mapping theorem. There are beautiful results such as the shear construction of Clunie and Sheil-Small [C-SS], and the theorem of Radó-Kneser-Choquet for convex harmonic mappings [K], [Ch]. The literature nevertheless appears to contain few results about such basic issues as the relation between the univalence of f and of h. In this paper we determine conditions under which the univalence of one of them implies that of the other. We also find conditions under which the harmonic mappings F = h+ eg remain univalent for all θ ∈ [0, 2π]. A domain Ω ⊂ C is linearly connected if there exists a constant M < ∞ such that any two points w1, w2 ∈ Ω are joined by a path γ ⊂ Ω of length l(γ) ≤M |w1−w2|, or equivalently (see [P]), diam(γ) ≤M |w1 −w2|. Such a domain is necessarily a Jordan domain, and for piecewise smoothly bounded domains, linear connectivity is equivalent to the boundary’s having no inward-pointing cusps. Our first result can be considered a harmonic version of [A-B]. Theorem 1: Let h : D → C be a holomorphic univalent map. Then there exits c > 0 such that every harmonic mapping f = h + g with dilatation |ω| < c is univalent if and only if h(D) is a linearly connected domain. The proof will show that c may be taken equal to 1 when h is convex, and we will show that c = 1 only in this case. Proof: Suppose first that Ω = h(D) is linearly connected, and let f = h+g be a harmonic mapping with |ω| < 1/M . We claim that f is univalent in D, or equivalently, that w + φ(w) is univalent in Ω, where φ = g ◦ h satisfies |φ| < 1/M . If not, then for w1 6= w2 we will have φ(w2)− φ(w1) = w1 − w2 . (1) ∗The authors were partially supported by Fondecyt Grant # 1030589.