21 problems

De Giorgi's conjecture. Any initial $n$-dimensional smooth submanifold $\varphi:M\to\mathbb{R}^{n+m}$ evolves by the gradient flow of $\mathcal{G}_k$ without developing singulariti…

In de Sitter space, let $M$ be a complete spacelike hypersurface with constant mean curvature. Goddard's conjecture. $M$ must be totally umbilic. This is a conjecture in Lorentzian…

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…

Higher-regularity homogeneity conjecture. The analogue of the theorem holds in the $C^r$ category for every $r\ge2$ and in the analytic category.

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 $\Sigma\subset\mathbb{R}^{d+n}$ be a $d$-dimensional submanifold, and let $\mathcal{T}(\Sigma)$ be the right-angled trapezoid of exponents $(1/p,1/q)$ for which the correspondi…

A spin field $cpsi$ satisfying the Killing spin field equation is considered, with products of previously constructed solutions and Clifford-algebra rotations as the proposed build…

Let $M$ be a surface of codimension two in $\mathbb{R}^{4}$ satisfying the pinching condition … The Baker–Nguyen conjecture. The Clifford torus is the true obstruction to the conve…

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 $M$) be a complete complex $n$-dimensional holomorphically immersed submanifold of $d53f^{n+r}$, and let $\sigma$ denote its second fundamental form. The quantity $|\sigma|^2$…

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…

Conjecture $-E$. Every compact smooth manifold $Y^n$ admits a smooth embedding into $S^{2n}$ such that all principal curvatures of the image satisfy

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 Riemannian manifold, let $p\in M$, let $W\subset T_pM$ be a linear subspace, and let $v$ be a vector orthogonal to $W$. A submanifold $S$ is a submanifold with prescri…

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.

Let $M$ be a complex equifocal submanifold, and suppose that the following condition holds: the complex focal set of $M$ at any point $x\in M$ consists of infinitely many complex h…

Sharp generalized Reilly constant conjecture. The constant $C_R(n,k)$ is sharply bounded by a constant times $k^{2/n}$ when $q=0$; when $q\ne 0$, it is correspondingly bounded by a…

Let $M$ be a submanifold as in the generalized Reilly inequality, with dimension $n$, eigenvalue index $k$, potential $q$, and generalized Reilly constant … For $q=0$, the constant…