In very sketchy terms, one standard proof of the Donaldson-Uhlenbeck-Yau theorem (closest to Simpson’s approach [72]) proceeds via the heat flow. It has two principal steps:

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  Consider the HYM heat flow

  $$\partial_{t}HH^{-1}=-\sqrt{-1}\Lambda F_{H}.$$

  Using certain parabolic maximum principles, one proves long time existence by showing that all derivatives of $H$ remain bounded for any given finite time. Furthermore, $\sup_{X^{\vee}}|\Lambda F|$ and $\left\lVert F\right\rVert_{L^{2}}$ are non-increasing in time, so remain uniformly bounded for all time. This step does not use stability.

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  The Donaldson functional is non-increasing in time almost by definition, so has a uniform upper bound for all time. Together with the pointwise bound on $\Lambda F$, which is like a Laplacian bound, one eventually shows that if $H(t)$ fails to be $L^{\infty}$ bounded for all time, then there exists an $L^{2}$-subbundle of $E$ with destabilizing properties, which is then interpreted algebraically as a subsheaf. (Roughly, the destabilizing sheaf comes from the eigensubspaces of $E$ corresponding to the small eigenvalues of $H(t)$ with respect to a fixed reference metric; compare the variational viewpoint below.) The stability condition rules out this case; one then shows that $H(t)$ actually converges smoothly at infinite time to a solution of the HYM equation.

We now discuss the variational perspective to the HYM equation, even though no proofs have been constructed along such lines. One would try to compactify $\mathcal{H}$ in some weaker topology (which is not known),²⁵²⁵ 25 What is known is how to compactify the space of Kähler potentials via psh functions, see Boucksom [11] for its fantastic application to Kähler-Einstein metrics. In that context, the boundary at infinity is related to non-archimedean geometry. extend the Donaldson functional to this compactification, attempt to find a minimizer of the functional, and then prove its regularity. Since the Donaldson functional is convex, it is natural to expect the existence of minimizer is equivalent to the properness of $\mathcal{M}$, or roughly equivalently $\mathcal{M}$ should grow at the infinity of $\mathcal{H}$.

Suppose now that $E$ fits into an extension sequence

$$0\to E_{1}\to E\to E_{2}\to 0.$$

Take arbitrary Hermitian metrics $H_{E}$ and $H_{E_{2}}$ on $E$ and $E_{2}$ respectively,²⁶²⁶ 26 The fact that the choice of Hermitian metrics will not ultimately matter is an expected feature, analogous to the relation between Kähler potentials and non-archimedean potential theory. and regard $H_{E_{2}}$ as a semi-Hermitian metric on $E$. We can then produce a 1-parameter family of Hermitian metrics on $E$, via $H(s)=e^{-s}H_{E}+H_{E_{2}}$. For $s\gg 1$, the metric $H(s)$ can be understood as equal to $e^{-s}H_{E}$ when restricted to $E_{1}$, and almost equal to $H_{E_{2}}$ on the orthogonal complement of $E_{1}$. In terms of bundles with connections, in the limit $s\to\infty$ we get $E_{1}\oplus E_{2}$ with the Chern connection for $H_{E}|_{E_{1}}\oplus H_{E_{2}}$. As such, it is an easy exercise to show that to leading order

$$\mathcal{M}(H(s))\sim-2\pi s\int_{X^{\vee}}c_{1}(E_{1})\wedge\omega_{X^{\vee}}^{n-1}.$$

Recall we have assumed $c_{1}(E)=0$ to simplify the definition of the Donaldson functional. Thus the subbundle $E_{1}$ destabilizes $E$, precisely when the degree of $E_{1}$ is positive, so $\mathcal{M}(H(s))$ goes to $-\infty$ along this 1-parameter family.

If one wants to turn the variational approach into an actual proof, one needs to further show that all possible ways to approach the infinity of (the metric completion of) $\mathcal{H}$ can be approximated by such 1-parameter families of algebraic origin. For our motivational purpose, it suffices to emphasize the following conceptual points:

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  The interesting limiting behaviour of the Donaldson functional occurs at an infinite distance boundary of (the metric completion of) $\mathcal{H}$. This sits well with the non-positive Riemannian curvature of $\mathcal{H}$.

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  The stability condition controls the asymptotic behaviour of the Donaldson functional near the boundary of $\mathcal{H}$.