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

+ 4 like - 0 dislike

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.

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 in Mathematics by Siqi He (20 points) [ revision history ]
edited Jan 14, 2016 by dimension10.admin

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