19 problems

Schläfli's conjecture. The system is locally solvable provided that the target Euclidean space has dimension $d\geq n_{\ast}$. This concerns the existence of local isometric immers…

Borisov's conjecture. The Nash–Kuiper theorem should extend to $\C^{1,\alpha}$-regularity whenever

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

Nonexistence conjecture. No such isometric immersion is possible for any $\tilde{c}>-1$.

Schläfli–Yau conjecture. Every two-dimensional Riemannian manifold has a local isometric immersion into $\mathbb{R}^{3}$.

Minimal immersion conjecture. The immersion $f$ must be totally geodesic.

Let $M^{m}(c)$ be the $m$-dimensional space form of constant curvature $c$, and let $\widetilde{M}^{2m-1}(\widetilde{c})$ be the $(2m-1)$-dimensional space form of constant curvatu…

Let ${\mathbb H}^n$ be the $n$-dimensional hyperbolic space and let ${\mathbb E}^{2n-1}$ be Euclidean space of dimension $2n-1$. A global isometric immersion is an isometric immers…

Asymptotically Euclidean immersion conjecture. There exists an asymptotically Euclidean isometric immersion

Let $(\mathcal{S},\mathfrak{s})$ be a closed Riemannian manifold and let $\mathcal{M}\subset\mathcal{S}$ be a Lipschitz subdomain. Define the relaxed quantitative rigidity property…

The paper considers the Besov-type borderline space $b^{2/3}_{3,\infty}$, which is larger than the fractional Sobolev space $W^{2/3,3}$. Borderline-space conjecture. The results of…

Piecewise smooth immersion conjecture. There exists a piecewise smooth isometric immersion

Let $M^{n_1}_1$ and $M^{n_2}_2$ be Riemannian manifolds. Suppose that $M^{n_1}_1$ can be locally isometrically immersed in $\mathbb{R}^{n_1+p_1}$ but in no lower-dimensional Euclid…

Let $M$ be an $n$-dimensional manifold, let $s_n=n(n+1)/2$, and let ${\cal D}_{M,q}:\operatorname{Imm}^1(M,\mathbb{R}^q)\to{\cal G}^0(M)$ be the metric-inducing operator defined by…

Let $f:M^n\to\mathbb{R}^{2n-1}$ be an isometric immersion of a compact Riemannian manifold $M^n$. The extrinsic curvature condition in Lemma asserts that such an immersion cannot e…

Let $D=A+B$ be the class in the homology of $\mb{S}^2\times\mb{S}^2$, and consider the $8$-parameter family of complex tori representing $2D=2(A+B)$. For an immersed surface, let t…

Let $w$ be a map satisfying the hypotheses of Proposition, and suppose in addition that $w=Du$ for some scalar function $u$. Gradient regularity conjecture. The conclusions of Prop…

Let $m$ and $q$ be positive integers, and let ${\cal D}_{m,q}$ denote the operator … from $C^\infty(\mathbb R^m,\mathbb R^q)$ to the space of Riemannian metrics on $\mathbb R^m$, w…

Conjecture. Such a manifold is a bounded symmetric domain of rank one. This means, in particular, that it is a product of complex hyperbolic spaces.