We adapt the notions of stability of holomorphic vector bundles in the senseof Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vectorbundles for canonically polarized framed manifolds, i.e. compact complexmanifolds X together with a smooth divisor D such that K_X \otimes [D] isample. It turns out that the degree of a torsion-free coherent sheaf on X withrespect to the polarization K_X \otimes [D] coincides with the degree withrespect to the complete K\"ahler-Einstein metric g_{X \setminus D} on X\setminus D. For stable holomorphic vector bundles, we prove the existence of aHermitian-Einstein metric with respect to g_{X \setminus D} and also theuniqueness in an adapted sense.
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