I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14)

https://arxiv.org/pdf/1402.4203.pdf

For completeness, the statement is as follows.

**Theorem:** Let $M$ be a closed manifold and $\rho: \pi_1(M)\to SL_n(\mathbb{C})$ be a semisimple representation. Then there is a $\rho$-equivariant harmonic map from $\tilde{M}\to SL_n(\mathbb{C})/SU_n(\mathbb{C})$.

(Look at the survey starting from page 27, for the notations and conventions). The proof uses the well-known heat flow method, pioneered by Eells and Sampson. Namely, one considers the equation $$\frac{\partial}{\partial t} u_t = \tau(u_t)$$ where $\tau(\cdot)$ denotes the tension fields, $t\in [0,\infty)$, and $u_0$ is some fixed $\rho$-equivariant map. Note that if $u_t$ is connected to $u_0$ along the heat flow then $u_t$ is necessarily $\rho$-equivariant as well. One notes you can solve this equation for all times and there is a limiting map that is necessarily harmonic.

Here is the source of my confusion.

Eells-Sampson prove that given a map $f$ from a compact manifold to some other manifold, there is a short-time solution to the heat equation. They then give some sufficient conditions for long time existence. In particular, if this solution remains bounded and the target manifold has negative curvature, then we get long time existence and convergence to a harmonic map. Hamilton proves a similar statement but for maps between compact manifolds with boundary.

However, Wentworth is working on some cover of $M$, not necessarily compact. To obtain long time existence, namely a solution, Wentworth quotes the papers of Eells-Sampson, and Hamilton. He then shows the solution remains bounded and seems to apply some results from Eells-Sampson. I am wondering, why is he able to use these results? There must be something about the $\rho$-equivariance that is making this possible, but I'm not seeing it. (A similar thing is done in, say, Donaldson's original paper, and the same issue arises).

Thanks.