Univalence criteria for lifts of harmonic mappings to minimal surfaces

Abstract

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformal metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.2000 Mathematics Subject Classification. Primary 30C99; Secondary 31A05, 53A10. Key words and phrases. Harmonic mapping, Schwarzian derivative, curvature, minimal surface. The authors were supported in part by FONDECYT Grant # 1030589. lift to a minimal surface, and it is this lift whose univalence is implied by bounds on the Schwarzian derivative. For the underlying harmonic mappings, univalent or not, our criterion is also shown to imply estimates on distortion and properties of boundary regularity that are better than those known or conjectured (see [20] or [9]) for the full normalized class of univalent harmonic mappings. In this respect our investigation can be viewed as a harmonic analogue of earlier work on analytic functions by Gehring and Pommerenke [11] and Chuaqui and Osgood [6], [7], [8].A planar harmonic mapping is a complex-valued harmonic function f (z), z = x + iy, defined on some domain Ω ⊂ C. If Ω is simply connected, the mapping has a canonical decomposition f = h + g, where h and g are analytic in Ω and g(z 0 ) = 0 for some specified point z 0 ∈ Ω. The mapping f is locally univalent if and only if its Jacobian |h ′ | 2 − |g ′ | 2 does not vanish. It is said to be orientation-preserving if |h ′ (z)| > |g ′ (z)| in Ω, or equivalently if h ′ (z) = 0 and the dilatation ω = g ′ /h ′ has the property |ω(z)| < 1 in Ω.According to the Weierstrass-Enneper formulas, a harmonic mapping f = h + g with |h ′ (z)| + |g ′ (z)| = 0 lifts locally to map into a minimal surface, Σ, described by conformal parameters if and only if its dilatation ω = q 2 , the square of a meromorphic function q. The Cartesian coordinates (U, V, W ) of the surface are then given byWe use the notation f (z) = U(z), V (z), W (z)for the lifted mapping of Ω into Σ. The height of the surface can be expressed more symmetrically as

Schwarzian derivative criteria for valence of analytic and harmonic mappings

“…In [4] we generalized Nehari's univalence criterion to the (conformal) lift f of a harmonic mapping f to its associated minimal surface Σ. Under an inequality of the form…”

“…In earlier work [4] we obtained the following criterion for the lift of a harmonic mapping to be univalent. 8…”

The Schwarzian derivatives of harmonic functions and univalence conditions

“…In particular, the authors of [5] observed a deep connection of S f with lifts of harmonic functions onto minimal surfaces. In [15,16] some estimations of S f in some subclasses of univalent harmonic functions were obtained and many properties of the Schwarzian were established.…”

John surfaces associated with a class of harmonic mappings

“…On the other hand, a generalization of the univalence criteria of Nehari (1) is proven in [4]. In that article is shown that, if f = h +ḡ is harmonic in D, its dilatation is the square of some meromorphic function, and…”