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

+ 4 like - 0 dislike
755 views

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

This post imported from StackExchange MathOverflow at 2015-04-09 12:16 (UTC), posted by SE-user Siqi He

asked Apr 2, 2015
edited Jan 14, 2016

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOver$\varnothing$lowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). Please complete the anti-spam verification