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

2009/12/06 by Ye‐Lin Ou, Ou, Ye-Lin · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0912.1141

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, 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.

Citations

• Harmonic Mappings of Riemannian Manifolds

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• k-harmonic maps into a Riemannian manifold with constant sectional curvature

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