Abstract.
Let M m and N be two compact Riemannian manifolds without boundary. We consider the L 2 gradient flow for the energy \(\int_M\vert\Delta u\vert^2\). If \(m \leq 4\) and N has nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.
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Received: 27 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004
Mathematics Subject Classification (2000):
58E20, 58J35
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Lamm, T. Biharmonic map heat flow into manifolds of nonpositive curvature. Calc. Var. 22, 421–445 (2004). https://doi.org/10.1007/s00526-004-0283-8
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DOI: https://doi.org/10.1007/s00526-004-0283-8