In several PDE involving metrics (like the Hermite-Einstein equation for vector bundles and the constant scalar curvature Kahler equation for manifolds) there is a notion along these lines - If a solution $h$ to the PDE exists, then it is approximated by a sequence $h_k$ of "balanced metrics". In all the cases I know, the "correct notion" of balanced metrics comes from either some infinite-dimensional construction or the finite-dimensional Kempf-Ness theorem, i.e., choose some appropriate group action (usually one of the classical groups) on an appropriate finite dimensional manifold (usually the space of holomorphic embeddings into projective space or the Grassmannian) equipped with a natural line bundle (The Quillen determinant associated to some universal bundle which can be a virtual bundle too) and find the moment map.
However, the choices of the "appropriate" finite-dimensional objects seems rather ad hoc. So my question is "Suppose I identify a PDE as the zero locus of an infinite-dimensional moment map along with the line bundle whose curvature is the infinite-dimensional symplectic form, then is there a systematic way for me to define balanced metrics? (perhaps by infinite dimensional geometric quantisation or something maybe)" Another closely related question is "If yes, then is there a systematic way to find out the finite-dimensional Kempf-Ness interpretation of the said balanced metrics ?"