2012 # Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces

**Abstract:** 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), biharmonic maps between spheres (Theorem 2.6) and into spheres (Theorem 2.7) 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 conjec…

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“…Only partial answers to Chen's conjecture have been obtained for more than three decades, e.g. [1], [2], [10], [32]. In the case of hypersurfaces, Chen's conjecture is true for the following special cases:…”

confidence: 99%

“…Only partial answers to Chen's conjecture have been obtained for more than three decades, e.g. [1], [2], [10], [32]. In the case of hypersurfaces, Chen's conjecture is true for the following special cases:…”

confidence: 99%

“…Since 2000, biharmonic maps have been receiving a growing attention and have become a popular subject of study with many progresses. For some recent geometric study of general biharmonic maps see [BK], [BFO2], [BMO2], [MO], [NUG], [Ou1], [Ou4], [OL], [Oua] and the references therein. For some recent study of biharmonic submanifolds see [CI], [Ji2], [Ji3], [Di], [CMO1], [CMO2], [BMO1], [BMO3], [Ou3], [OT], [OW], [NU], [TO], * [CM], [AGR] and the references therein.…”

mentioning

confidence: 99%

“…It is well known that any hypersurface is locally the graph of a function. As minimal graphs have play an important role in the study of minimal (hyper)surfaces, the author initiated the study of biharmonic graphs in [66]. where the Laplacian ∆ and the gradient ∇ are taken with respect to the induced metric g ij = δ ij + f i f j .…”

confidence: 99%