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

By Ye-lin Ou. 2009

56 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5fbfd183dd695eb5be55cc77fa4ef33aed9b1a90

How researchers see this paper

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In Applied Mathematics

1. Marian Ioan Munteanu
2. Bang‐Yen Chen
3. S. Maeta
4. Ion Mihai
5. Bayram Sahin
6. I. Hasegawa
7. Siraj Uddin

In Geometry and Topology

1. Bang‐Yen Chen
2. Marian Ioan Munteanu
3. S. Maeta
4. Ion Mihai
5. I. Hasegawa

Citations in the corpus

• Biharmonic ideal hypersurfaces in Euclidean spaces (2013, background): “In particular, there exist many results on the generalized Chen’s conjecture (see, for instance, [1–4,26–30]).”
• k-harmonic maps into a Riemannian manifold with constant sectional curvature (2010, background): “There are several results for this conjecture ([8], [5] and [10] etc).”