Some of the most significant results on the dHYM equation are:

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  There is a well developed formal GIT picture [23, section 2][19][21], including a complexified group action on an infinite dimensional Kähler manifold, a moment map, a negatively curved symmetric space, and a Donaldson type functional which is convex along geodesics.

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  The geodesic equation with prescribed boundary data have $C^{1,1}$-solutions and viscosity solutions under a large phase assumption [18, Thm 2.7]. This leads to nontrivial algebraic obstructions [18, section 3].

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  Assume large phase $\hat{\theta}>\frac{(n-2)\pi}{2}$ and the existence of a subsolution in the suitable sense, then the dHYM equation can be solved [18, Thm 5.2].

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  Assume large phase $\hat{\theta}>\frac{(n-2)\pi}{2}$, then the existence of dHYM solution can be formulated in terms of algebraic obstruction conditions [16].