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We give several construction methods and use them to produce many examples ofproper biharmonic maps including biharmonic tori of any dimension in Euclideanspheres (Theorem 2.2, Corollaries 2.3, 2.4, and 2.6), biharmonic maps betweenspheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonalmultiplications and eigenmaps. We also study biharmonic graphs of maps, derivethe equation for a function whose graph is a biharmonic hypersurface in aEuclidean space, and give an equivalent formulation of Chen's conjecture onbiharmonic hypersurfaces by using the biahrmonic graph equation (Theorem 4.1)which paves a way for analytic study of the conjecture.

Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces