Given two closed Riemannian manifolds M of dimension 2m and N & SUB; RK, we study the existence problem of extrinsic mpolyharmonic maps in a fixed free homotopy class from M to N. We prove that any energy-minimizing sequence in a fixed free homotopy class subsequently converges locally in Wm,2 to a smooth extrinsic m-polyharmonic map except possibly at most finitely many energy concentration points. Moreover, at an energy concentration point we show that there exists a non-constant smooth extrinsic m-polyharmonic map from R2m to N by blow-up analysis. As a consequence, when the homotopy group ir2m(N) is trivial, we prove that there always exists a minimizing extrinsic m-polyharmonic map in every free homotopy class in [M2m, N]. This generalizes the celebrated existence results for harmonic maps (m = 1) and biharmonic maps (m = 2). The main technical ingredient is an & epsilon;-regularity for an energy-minimizing sequence, which is new for m-polyharmonic maps and should be of independent interest.
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