30 problems

Let $M^n$ be an $n$-dimensional manifold isometrically immersed in the space form $N^{n+m}(c)$ of constant sectional curvature $c$. Let $h$ be the second fundamental form, let…

Let $M^n$ be a closed, minimally immersed submanifold in the unit sphere $\mathbb{S}^{n+m}$ with constant scalar curvature, equivalently with constant length of the second fundamen…

Let $M$ be a biharmonic submanifold of a Euclidean space. Chen's conjecture. Every such submanifold is minimal. Chen, Ishikawa, and Jiang proved the claim for biharmonic surfaces i…

Let $M$ be a biharmonic submanifold of a non-positively curved Riemannian manifold. Generalized Chen's conjecture. The submanifold $M$ is minimal. Ou and Tang constructed counterex…

DDVV conjecture.

Let $M$ be a biharmonic submanifold of a sphere. BMO conjecture. The submanifold $M$ has constant mean curvature. The source presents this as one of the well-known open conjectures…

Let $M^n$ be a closed minimal submanifold of the unit sphere $\mathbb S^{n+m}$. With respect to a local orthonormal normal frame, let $\mathcal A$ be the fundamental matrix of the…

Let $M^n$ be an immersed submanifold of the space form $N^{n+m}(c)$, with scalar curvature $\rho$, normal scalar curvature $\rho^{\perp}$, and mean curvature tensor…

Leung's stronger conjecture. If $n$ is odd and $\sigma(W)<\frac{n+1}{n-1}$, then $W$ is homeomorphic to $S^n$.

Let $M_{0}$ be an $n$-dimensional complete submanifold in the sphere $\mathbb{S}^{n+q}\left(1/\sqrt{c}\right)$. Let $h$ be its second fundamental form, $H$ its mean curvature vecto…

Let $M$ be a complete submanifold of Euclidean space $\mathbb{E}^n$. An isometric immersion $\phi:(M,g)\rightarrow\mathbb{E}^n$ is biharmonic when its bitension field satisfies…

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 $M\hookrightarrow \mathbb{C} P^N$ be a complete, connected, and full complex submanifold, meaning that it is not contained in a proper hyperplane. Suppose its normal holonomy g…

Classification conjecture. Every closed totally magnetic submanifold of $E(\mathbf a)$ is of the form

Let $\varphi:M^2\to N^4(\epsilon)$ be a proper biharmonic immersion. Generalized Chen conjecture. Then $\epsilon>0$, so $N^4(\epsilon)$ is the four-dimensional sphere…

Let $\varphi:M^2\to N^4(\epsilon)$ be a non-CMC W-surface with flat normal bundle. Assume that $3f^2+K-\epsilon\neq 0$ and $\nabla^\perp E_3\neq 0$ at any point. Biharmonicity conj…

Existence and uniqueness conjecture. There exists an isometric immersion $f:M\to(\mathbb{C}P^3,g,J)$ and a vector bundle isometry $\varphi:\mathcal{E}\to T^\perp M$ such that…

Let $M^n$ be an immersed submanifold of a real space form with constant sectional curvature $\kappa$. The normalized scalar curvature is denoted by $\rho$, the normalized normal sc…

Lei–Xu's conjecture. If

Spherical embedding conjecture. If, in addition, $M$ satisfies

Leung's conjecture. If $n$ is odd and $\sigma(W)\leq\frac{n}{n-1}$, then $W$ is homeomorphic to $S^n$.

Let $M$ be a compact manifold with positive curvature, and let $\cal F$ be a Riemannian foliation on $M$. A horizontal vector is a vector tangent to the horizontal distribution of…

Let $F_{0}:M^{n}\to\mathbb{S}^{n+q}$ be an $n$-dimensional closed submanifold satisfying … where $h$ is the second fundamental form. Lawson–Simons conjecture. The mean curvature fl…

Leung's weak pinching conjecture. If $n$ is odd and $\sigma(W)\leq\frac{n}{n-1}$, then $W$ is homeomorphic to $S^n$.

Let $M^k$ be an irreducible Wintgen ideal submanifold of dimension $k\ge 3$, meaning that the only integrable distribution containing the canonical distribution $\mathbb{D}_2$ is t…