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Consider $E$ a holomorphic vector bundle over a projective manifold $X$polarized by an ample line bundle $L$. Fix $k$ large enough, the holomorphicsections $H^0(E\otimes L^k)$ provide embeddings of $X$ in a Grassmanian space.We define the \textit{balancing flow for bundles} as a flow on the space ofprojectively equivalent embeddings of $X$. This flow can be seen as a flow ofalgebraic type hermitian metrics on $E$. At the quantum limit $k\to \infty$, weprove the convergence of the balancing flow towards the Donaldson heat flow, upto a conformal change. As a by-product, we obtain a numerical scheme toapproximate the Yang-Mills flow in that context.

Quantization of Donaldson's heat flow over projective manifolds