20 problems

Banach's isometric embedding conjecture. $L_p$ isometrically embeds into $L_r$.

Gromov's flexibility threshold conjecture. The threshold for flexibility in codimension one is

Intrinsic characterization conjecture. $(\mathbb S^n,g)$ admits a smooth global isometric embedding

The Hölder exponent $\alpha$ measures the regularity of a $C^{1,\alpha}$ isometric embedding, and the preceding flexibility and rigidity results leave the precise threshold between…

Let $s_d=d(d+1)/2$ be the Janet dimension, and let a $d$-dimensional smooth Riemannian manifold be given. A smooth local isometric embedding is a smooth map that locally preserves…

Let $(\mathcal M,g)$ be an $n$-dimensional compact smooth Riemannian manifold, and consider $C^{1,\theta}$ isometric embeddings of $(\mathcal M,g)$ into Euclidean space. The known…

Let $T$ be a finite metric tree, and let $n$ be a positive integer. The tree $T$ has at most $2^n$ leaves. An isometric embedding of $T$ into $(\mathbb{R}^n,d_{\infty})$ is a map p…

For $p,q>0$, let $(\mathbb C^2,\|\cdot\|_{B_{p,q}})$ denote the two-dimensional Banach space equipped with the $B_{p,q}$ norm, and let $S_\infty$ be the space of compact operators…

Let $h$ be a sufficiently smooth Riemannian metric on the unit sphere $\mathbb{S}^2$ with positive Gauss curvature $K_h$. Weyl's conjecture. The metric $h$ may be realised as a con…

Let $X$ be a normed space with unit sphere $S_X$, and let $Y$ range over normed spaces with unit spheres $S_Y$. An isometric embedding $S_Y\to S_X$ is a distance-preserving map bet…

Let $n$ be a positive integer, and let $\\|\cdot\\|_X$ and $\\|\cdot\\|_Y$ be normalized norms on $\mathbb R^n$, meaning that $\\|e_i\\|_X=\\|e_i\\|_Y=1$ for every standard basis v…

Let $\Sigma$ be a $2$-dimensional compact Riemannian manifold, possibly with boundary, with positive Gauss curvature. A $C^{1,\alpha}$ isometric embedding is an isometric embedding…

Let $\Sigma_a$ be the polar cap and let $\mathscr{I}^\alpha_1(\Sigma_a)$ denote the class of $C^{1,\alpha}$ isometric embeddings of $\Sigma_a$ into the relevant codimension-one Euc…

Let $W_4$ denote the wheel on $5$ vertices. For a graph $G$, let $f_\infty(G)$ be the least dimension $k$ such that every distance function $d:E(G)\to\mathbb{R}_{\ge 0}$ admits vec…

Let $M\subset\mathbb{R}^3$ be a complete embedded minimal surface, and let an intrinsic isometry of $M$ or another minimal embedding of $M$ into $\mathbb{R}^3$ be given. Isometric…

Nash's conjecture. The embedding $f$ should be $\epsilon$-close to a $C^1$ isometric embedding for every $\epsilon>0$ whenever $N\geq n+1$.

Bryant–Griffiths–Yang conjecture. The characteristic variety $\Sigma({\bf c})$ is not smooth in $\mathbb R^n\setminus\\{0\\}$ for every $n\ge 5$.

Kalton's conjecture. Such an $X$ should embed almost isometrically into an $\ell_p$-sum of finite-dimensional spaces.

Let $n\geq 3$ and let $M^n$ be an $n$-dimensional $PL$-manifold. A PL isometric embedding is an embedding of $M^n$ into Euclidean space that preserves its piecewise-linear metric.…

Let $M^m$ be the manifold and let $\tilde f$ be the isometric embedding map constructed in Theorem 3.6. Homotopy uniqueness conjecture. The isometric embedding map $\tilde f$ in Th…