Chen's conjecture on biharmonic submanifolds in Riemannian manifolds

Keomkyo Seo; Gabjin Yun

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arXiv · 2108.10667

Chen's conjecture on biharmonic submanifolds in Riemannian manifolds

Keomkyo Seo·Gabjin Yun

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Abstract

We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian manifold. One of interesting problems in this direction is Chen's conjecture which says that any biharmonic submanifold in a Euclidean space is minimal. From the invariant equation for biharmonic submanifolds, we derive a fundamental identity involving the mean curvature vector field, and using this, we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space. More generally, it is proved that any biharmonic submanifold in a space form of nonpositively sectional curvature is minimal. Furthermore we provide affirmative partial answers to the generalized Chen's conjecture and Balmu\c{s}-Montaldo-Oniciuc conjecture.

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BibTeX RIS

Keomkyo Seo, Gabjin Yun. 2021-08-24. Chen's conjecture on biharmonic submanifolds in Riemannian manifolds. https://arxiv.org/abs/2108.10667

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