The reverse direction, that $\mu$-stability implies the existence of HYM connections, is the hard part of the subject, and a key ingredient is the Donaldson functional. We make the not very essential simplification that $c_{1}(E)=0$. Donaldson [26][27] defined a functional $\mathcal{M}$ on the infinite dimensional space $\mathcal{H}$ of Hermitian metrics on the fixed bundle $E$, by prescribing its first variation at any point $H\in\mathcal{H}$:

$$\delta\mathcal{M}=\int_{X^{\vee}}\Tr(\sqrt{-1}F_{H}\delta HH^{-1})\wedge\omega_{X^{\vee}}^{n-1}.$$

(8)

Obviously from the definition, the critical points are precisely the HYM metrics. Less obviously, this functional is well defined up to an additive constant fixed by the choice of a reference Hermitian metric $H_{0}$. Different choices are related by

$$\mathcal{M}_{H_{0}}(H)=\mathcal{M}_{H_{0}^{\prime}}(H)+\mathcal{M}_{H_{0}}(H_{0}^{\prime}).$$

(9)

The space $\mathcal{H}$ can be formally assigned a Riemannian structure with non-positive curvature:

$$|\delta H|^{2}=\int_{X^{\vee}}\Tr(\delta HH^{-1})^{2}\wedge\omega_{X^{\vee}}^{n}.$$

(10)

The geodesics in $\mathcal{H}$ are given by $e^{tA}H$ where $A$ is self adjoint with respect to $H$, and the Donaldson functional is convex along geodesics.