9 problems

A biharmonic submanifold is a submanifold whose inclusion map is biharmonic; a minimal submanifold has vanishing mean curvature. Let a submanifold lie in a Riemannian manifold whos…

A proper biharmonic submanifold is a biharmonic submanifold that is not minimal. A submanifold has constant mean curvature when the length of its mean-curvature vector is constant.…

Let $\varphi:M^m\hookrightarrow\mathbb{S}^{m+1}$ be a biharmonic submanifold, and let $\mathbf H$ denote its mean curvature vector field. The mean curvature has constant length whe…

Let $M$ be a complete biharmonic submanifold of Euclidean space. Akutagawa–Maeta conjecture. $M$ is minimal. This is a global reformulation of Chen's conjecture for complete subman…

Classification conjecture. The only proper biharmonic hypersurfaces in $\mathbb{S}^{m+1}$ are the open parts of hyperspheres $\mathbb{S}^m(1/\sqrt{2})$ or of the standard products…

Let $\varphi:M^m\hookrightarrow\mathbb{S}^{m+1}$ be a biharmonic hypersurface, and let $\mathbf H$ denote its mean curvature vector. The hypersurface has constant mean curvature (C…

Let $\mathbb T=\mathbb S^1\left(\frac{1}{2}\right)\times\mathbb S^1\left(\frac{1}{2}\right)$ be the flat Clifford torus, and let $\Phi:\mathbb T\to\mathbb S^4$ be the proper biharm…

CMC conjecture. Any biharmonic submanifold in a Euclidean sphere is CMC.

Global Chen conjecture. Any complete biharmonic immersed submanifold in $\mathbb{E}^N$ is minimal.