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Let (M, g) be a four dimensional compact Riemannian manifold without boundary, (N, h) ⊂ R be a compact Riemannian submanifold without boundary. We establish the existence of a global weak solution to the heat flow of extrinsic biharmonic maps from M to N , which is smooth away from finitely many singular times. As a consequence, we prove that if Π4(N) = {0}, then any free homotopy class α ∈ [M, N ] contains at least one minimizing biharmonic map.

Heat Flow of Biharmonic Maps in Dimensions Four and Its Application