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*](http://arxiv.org/abs/math/9810033), 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



Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence
The Korevaar-Schoen limit of the $u\_k$ function is the following map


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


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 
\begin{equation}
\mathbb{R}^1\times S^2 \rightarrow \mathbb{R}^1\\
(t,x)\rightarrow t
\end{equation}


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


