Images of Harmonic Maps with Symmetry

We show that under certain symmetry, the images of complete harmonic embeddings from the complex plane into the hyperbolic plane is completely determined by the geometric information of the vertical measured foliation and is independent of the horizontal measured foliation of the corresponding Hopf differentials. In this paper, we find a new explicit relation between the image of harmonic embeddings, with certain symmetry, from the complex plane C into the hyperbolic plane H and the metric of the associated R-tree of the corresponding vertical measured foliation of the Hopf differentials. Unlike in the case of compact surfaces, holomorphic quadratic differentials cannot be determined by the vertical measured foliation only. So it is kind of surprising for us to find that the image set of the corresponding complete harmonic embedding is completely determined by the vertical measured foliation and is independent of the geometric information of the horizontal measured foliation. The symmetry condition that we consider is as follow. We assume that the harmonic embedding u from C into H is invariant under the group Zk by rotations and its image is an ideal polygon with 2k vertices for any integer k ≥ 2. This is the next nontrivial case after the case of Z2k symmetry which gives harmonic embeddings with regular polygonal images. This condition can be regarded as u having half of the symmetry of a regular polygon. The symmetry assumption implies that the Hopf differentials are of the form [z − (a + ib)zm−1]dz2 for a+ ib ∈ C. For a generic holomorphic quadratic differential in this family, the associated R-tree has m+1 finite edges of equal length given by ν = π|b|/(2(m+1)). We will show that Theorem 1. Let u : C → H be the unique (up to equivalence) complete orientation preserving harmonic embedding associated to a quadratic differential equivalent to [z2m−(a+ ib)zm−1]dz2. Then, up to isometry, the image u(C) is the interior of the ideal polygon with vertices given 2000 Mathematics Subject Classification. Primary 53C43. The second author is partially supported by Earmarked Grants of Hong Kong CUHK4291/00P. 1 2 THOMAS K. K. AU & TOM Y. H. WAN by {1, e, ω, ωe, . . . , ω, ωmeiα} in the unit disc model of H, where ω = e, α = αm(ν) = 2 tan −1 ( sin(π/(m+ 1)) cos(π/(m+ 1)) + e2ν ) , and ν = π|b|/(2(m + 1)) is the common length of the finite edges of the R-tree associated to the quadratic differential given by Lemma 1.1. In this paper, a harmonic embedding u is called complete if its ∂-energy metric ‖∂u‖2|dz|2 is a complete metric on C, where z is the standard complex coordinate on C. The result is related to the work of Shi and Tam [7]. The facts that complete harmonic embeddings from C to H are parametrized by Hopf differentials [8, 9] and the images are determined by the asymptotic behaviors of the harmonic embeddings [1, 2, 3], suggest the following problem as a step toward Schoen’s conjecture [6] on the nonexistence of harmonic diffeomorphism from the complex plane to the hyperbolic plane: Suppose that u is a complete orientation preserving harmonic embedding with polynomial Hopf differential P (z)dz, is it possible to find explicit relation between the coefficients of P (z) and the vertices of u(C)? For this problem, they showed that, up to isometry, the image of a complete orientation preserving harmonic embedding from the complex plane into the hyperbolic plane is a regular ideal polygon if its Hopf differential is given by (z2m−azm−1)dz2 for some real number a. This is the first nontrivial example of a family of harmonic maps (for fixed m) with identical images. It is obvious that our result is a generalization of that of Shi-Tam. However, the method is quite different. In [7], the authors studied the asymptotic behavior of the image of the harmonic maps along euclidean rays to infinity. Our approach adopts more geometric properties of the Hopf differential, especially those related to the metric information of the R-tree associated to the vertical measured foliation of the Hopf differential. The relationship between the asymptotic behavior of harmonic maps and the associated R-trees has been studied by Minsky [5] and Wolf [10, 11, 12], independently. In these works, the asymptotic behavior of a sequence of harmonic maps on a compact surface with energy (or the norm of the Hopf differential) going to infinity was studied. In our case, instead of a sequence of maps, we are interested in the asymptotic behavior of harmonic maps on a complete noncompact surface as in [3]. In particular, the asymptotic behavior of the length of the image of a horizontal trajectory near IMAGES OF HARMONIC MAPS 3 infinity was studied. More precisely, it was shown that the image of a horizontal trajectory is asymptotic to a geodesic; and the difference between the lengths of this image and the asymptotic geodesic is actually tending to zero as the Φ-distance is going to infinity. The arrangement of this paper is as follows. In Section 1, we will give a brief description of harmonic maps, its Hopf differentials and the geometric information of the R-trees associated to the Hopf differentials. Then we will study the asymptotic behavior of the image of horizontal trajectories in Section 2. Finally, we prove our main result in Section 3. 1. Background 1.1. Harmonic maps between surfaces. Let M and N be oriented surfaces with metrics ρ2|dz|2 and σ2|du|2, where z and u are local complex coordinates of M and N , respectively. A C map u from M to N is harmonic if and only if u satisfies uzz̄ + 2 (log σ(u))u uzuz̄ = 0. The Hopf differential Φ = φ(z)dz of a map u between these surfaces is defined by φ(z) = σ (u(z)) uz(z)ūz(z). If u is harmonic, then it is well-known that Φ is a holomorphic quadratic differential on M . The ∂-energy density and ∂-energy density of u are defined by ‖∂u‖ = σ (u) ρ2 |uz| and ‖∂u‖ = σ(u) ρ2 |uz̄|. In terms of the ∂-energy density and ∂-energy density, the energy density and Jacobian of u can be written as e(u) = ‖∂u‖ + ‖∂u‖ and J(u) = ‖∂u‖ − ‖∂u‖. In this paper, we are interested in the case that M = C, N = H, and that u : C −→ H is an orientation preserving open harmonic embedding. In this case, the Jacobian is strictly positive, i.e., J(u) > 0, and hence ‖∂u‖2 > 0. Therefore, one can consider the ∂-energy metric ‖∂u‖2|dz|2 on the complex plane C. As mentioned in the introduction, u is called complete if its ∂-energy metric ‖∂u‖2|dz|2 is a complete metric on C. As the completeness is only defined 4 THOMAS K. K. AU & TOM Y. H. WAN for orientation preserving u, the term complete harmonic open embedding implies implicitly that the harmonic embedding is orientation preserving. It was shown in [8, 9] that for each holomorphic quadratic differential Φ = φ(z)dz which is not identically zero, there is a complete harmonic open embedding, unique up to conformal transformations, u : C −→ H such that the Hopf differential of u is exactly Φ. 1.2. Trajectory structures and measured foliations of the Hopf differentials. Let Φ be a holomorphic quadratic differential on C, which is given in local coordinate z as Φ = φ(z) dz, where φ is in general a holomorphic function. For any z0 ∈ C with φ(z0) 6= 0, there is a choice of a continuous branch of √ φ(z) in a neighborhood W of z0. Then for a given base point z∗ ∈ W sufficiently close to z0, the mapping ζ(z) = ∫ z