Stable Higgs bundles and Hermitian-Einstein metrics on non-Kähler manifolds

Let $X$ be a compact Gauduchon manifold, and let $E$ and $V_0$ be holomorphic vector bundles over $X$. Suppose that $E$ is stable when considering all subsheaves preserved by a Higgs field $θ\in H^0($End$(E)\otimes V_0)$. Then a modified version of the Donaldson heat flow converges along a subsequence of times to a solution of a generalized Hermitian-Einstein equation, given by $iΛF+[θ,θ^\dagger]=λI$.

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