# Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence

Thank you for spending time on the following question.

I am trying to make an explicit example of Korevaar-Schoen convergence. The problem I am facing is that I cannot find the limit of the harmonic maps in the following example I hope to consider. The process I am doing mostly follows from [1].

Let $M = S^1\times S^2$, and consider the hyperbolic 3-space $X_k=\mathbb{H^3}=\{(x_1,x_2,x_3)\vert x_3>0\}$ with the canonical metric $$d_{\mathbb{H^3}}=\frac{dx_1^2+dx_2^2+dx_3^2}{x_3^2}$$ The representation $\rho_k: \pi_1(M) \rightarrow SL(2;\mathbb{C})$ is $$\rho_k(1) = \begin{bmatrix} k & 0 \\ 0 & \frac{1}{k} \end{bmatrix}$$ The action of $\rho_k(1)$ on $\mathbb{H}^3$ in the case we need to use is $$\rho_k(1)(0,0,x_3)=(0,0,k^2x_3)$$

By a theorem of Donaldson, we can find an equivariant map $u_k$ from the universal cover of $M$ to $\mathbb{H}^3$ satisfying the relation $u_k(\alpha x) = \rho_k(\alpha)u_k(x)$ ($\alpha$ here is an element in $\pi_1(M)$). In the case I am considering, we can construct the following harmonic map: \begin{align} u_k :& \mathbb{R}^1\times S^2 \longrightarrow \mathbb{H}^3 \\ &(t,x) \longmapsto (0,0,e^{C_kt}) \end{align} Here, $C_k = 2\ln k$ is a constant related to k. With this definition, $u_k$ is an equivariant map. The reason $u_k$ is harmonic is that the only harmonic map from $S^2$ to $\mathbb{H}^3$ is constant and the image of $u_k$ is geodesic. The energy I compute for this map is $E(u_k) = C_k^2$.

By theorem 3.1 of [1], if we define $\hat{d}_{\mathbb{H}^3,k} = d_{\mathbb{H}^3}$, we have a Korevaar-Schoen limit of the rescaled harmonic maps $u_k:\widetilde{M}\rightarrow (\mathbb{H^3},\hat{d}_{\mathbb{H}^3,k})$ to a map $\widetilde{M}\rightarrow T$, where $T$ is a $\mathbb{R}$-tree.

Question:
Here is my goal: I hope to know explicitly the limit in this case.

The problem I have is that I don't think the Korevaar-Schoen limit exists in the example I consider when I try to use the definition in [2]. Because $\frac{e^{C_kt}}{C_k^2}$ for a fixed $t$ will converge to infinity when $k$ goes to infinity. I don't know where I made a mistake. Maybe the maps $u_k$ I constructed are not harmonic?

[1] G. Daskalopoulous, S. Dostoglou, R. Wentworth, Character Varieties and Harmonic Maps to R-trees, arXiv:math/9810033v1

[2] N. Korevaar, R. Schoen, Global Existence Theorems for Harmonic maps to Non-locally compact spaces, Comm. Anal. Geom. 5(1997), no.2, 213-266

The Korevaar-Schoen limit of the $u_k$ function is the following map

$$\mathbb{R}^1\times S^2 \rightarrow \mathbb{R}^1\\ (t,x)\rightarrow (0,0,t)$$

The essential part of [1] is that we have the pointwise bound the energy function of a harmonic map.

Lemma[3]: Let $\Omega\subseteq M$ be a region with compact closure. For any compact set $K\subseteq \Omega$, there is a constant C depending only on $K, \Omega$, and the metric g of M such that for any complete manifold N of nonpositive curvature and any harmonic map $u:\Omega\rightarrow N$, we have the bound $sup_K e(u)\leq cE_{\Omega}(u)$. $e(u)$ is the energy density of u and E(u) is the energy of u.

Define the metric $\tilde{d_k}:=\frac{d_{\mathbb{H}^3}}{E(u_k)^{\frac{1}{2}}}$ and a metric $\hat{d_k}(x,y)=\tilde{d_k}(u_k(x),u_k(y))$, we get a metric on $\widetilde{M}$, Korevaar-Schoen Convergence is about the pointwise convergence of $\hat{d_k}$. The reason $\hat{d_k}$ exist a limit is that by using the lemma, we immediately know $\hat{d_k}(x,y)\leq d_{\widetilde{M}}(x,y)$(Which is shown in Theorem 3,1 of [1]). Therefore, the $\hat{d_k}(x,y)$ have a subsequence which is point wise convergence.

In the case $u_k(t,x)=(0,0,e^{C_kt})$, $\hat{d_k}((t_1,x_1),(t_2,x_2))=t_1-t_2$. When k goes to infinity,$\hat{d_k}$ convergence to the distance function $d_{\infty}((t_1,x_1),(t_2,x_2))=t_1-t_2$. Therefore, the Korevaar-Schoen limit of $u_k$ is $$\mathbb{R}^1\times S^2 \rightarrow \mathbb{R}^1\\ (t,x)\rightarrow t$$

Thanks Professor Dostoglou for his kindly reply of the important point of their paper.

[1] G. Daskalopoulous, S. Dostoglou, R. Wentworth, Character Varieties and Harmonic Maps to R-trees, arXiv:math/9810033v1

[2] N. Korevaar, R. Schoen, Global Existence Theorems for Harmonic maps to Non-locally compact spaces, Comm. Anal. Geom. 5(1997), no.2, 213-266

[3] R. Schoen, The Role of Harmonic Mappings in Rigidity and Deformation Problems. In Complex Geometry, 179-200, Lecture Notes in Pure and Applied Mathematics, Volume 143, Marcel Dekker, New York, 1993

• You should accept your answer in order that the question is marked as answered to satisfaction. – Benoît Kloeckner May 3 '15 at 20:30
• @BenoîtKloeckner Thank you for your comment. – Siqi He May 3 '15 at 21:21