9 problems

Let $M^{2n-1}\be2 {\bf R}^{2n}$ be a non-centrally symmetric hypersurface. A skew brane is an $(2n-1)$-dimensional submanifold of ${\bf R}^{2n+1}={\bf R}^{2n}\be2 {\bf R}$; the ver…

Let $M$ be an isoparametric hypersurface or a focal submanifold of the unit sphere $\mathbb{S}^n$. Tang's conjecture. Every such isoparametric hypersurface and focal submanifold ad…

Let $M$ be an $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ with the induced metric. Let $\sigma$ be its second fundamental form, let $\ast1$ be the volume form, and de…

Let $\mathcal{M}$ be a compact $m$-dimensional submanifold of $\mathbb{R}^n$ with codimension $k=n-m$ and proper curvature conditions. Let $N_{\mathcal{M}}(Q,\delta)$ count rationa…

Xu–Gu's conjecture. If

Let $M$ be a closed submanifold of Euclidean space, let $m=\dim M$, and let $E_M(z)$ denote its regularized Riesz energy. The scale-anomaly term $R_M(-2m)$ measures the failure of…

Let $M$ be a smooth closed $n$-dimensional submanifold, with $n\geq2$, in the space form $\mathbb{F}^{n+d}(c)$, where $c\geq0$. Let $\mathring{A}$ denote its traceless second funda…

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

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…