54 problems

A submanifold of Euclidean space is biharmonic when its inclusion map is a biharmonic map, equivalently a critical point of the bienergy functional. It is minimal when its mean cur…

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.…

DDVV conjecture. The normal scalar curvature conjecture asserts that the normal scalar curvature of $M$ satisfies the DDVV inequality.

Let $f:M^{n}\rightarrow \widetilde{M}^{m}(c)$ be an isometric immersion from $M^{n}$ into a real space form of constant curvature $c$. Let $\rho$ and $\rho^{\perp}$ denote the norm…

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 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…

Maeta's generalized Chen conjecture. Any $k$-harmonic submanifold in the Euclidean space $\mathbb R^{n+1}$ is minimal.

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…

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…

In a symmetric space of non-compact type, a complex equifocal submanifold with flat section is a submanifold satisfying the complex equifocality condition and having flat sections;…

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…

Global non-immersibility conjecture. For $R<1$, there is no global isometric immersion

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…

An isoparametric hypersurface with $g=4$ principal curvatures has multiplicities equal to those of a known Ferus–Karcher–Münzner example or one of the two homogeneous exceptions wi…

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

Classification conjecture. Every closed, connected totally magnetic submanifold of $E(A)$ of positive dimension is of the form

Minimality conjecture. The only $H$-tensional submanifolds of real space forms are the minimal ones.

Let $M^n$ be a closed immersed minimal submanifold of the unit sphere $\mathbb{S}^{n+q}$. Let $S_\alpha$ be the shape operators with respect to an orthonormal basis…

Let $M$ be a submanifold of a Riemannian manifold. An $HS$-tensional submanifold and an $HM$-tensional submanifold are the classes defined in the paper via the corresponding tensio…

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 $M^m$ be a submanifold of Euclidean space $\mathbb{R}^n$. An immersion is called $r$-harmonic if it is a critical point of the $r$-energy, and it is minimal when its mean curva…