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In Kronheimer [1, p.183], a certain statement is made of which I extract the following special case.

Let $\alpha:\mathbb{R}\to \mathrm{Mat}(n\times n,\mathbb{C})$ be smooth and suppose that there exist $C,a>0$ and a diagonal matrix $H$ such that $$\|\alpha(t)-H\|<Ce^{-at}$$ for all $t\in\mathbb{R}$. Then, there is a solution to the differential equation $$g(t)\alpha(t)g(t)^{-1}-\dot{g}(t)g(t)^{-1}=H,\quad(g(t)\in\mathrm{GL}(n,\mathbb{C})),$$ such that $$\lim_{t\to\infty}g(t)=1.$$

I see that this is just a first-order linear ODE and hence solutions exist globally, but I don't see why there must be a solution converging to $1$. There is no justification in the paper. How do we prove this?

Reference.

[1] Kronheimer, P., Instantons and the geometry of the nilpotent variety, J. Differential Geometry 32 (1990) 473-490.

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  • $\begingroup$ 1 means identity here ? $\endgroup$
    – Piyush Grover
    Commented Sep 20, 2017 at 13:28
  • $\begingroup$ @PiyushGrover Yes. $\endgroup$
    – Weisbrot
    Commented Sep 20, 2017 at 13:29
  • $\begingroup$ This should be easy if $a$ dominates the entries of $H$, and in general, I'm not even sure right now it's true (though it well might be). $\endgroup$
    – Christian Remling
    Commented Sep 20, 2017 at 18:30

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