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

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

2009-12-06

arxiv.org/abs/0912.1141v1

Mathematics

Differential Geometry

21 pages

Scientific paper

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

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