14 problems

Let $n\geqslant 2$, and let $M^n(c)$ be an $n$-dimensional model space of constant sectional curvature $c\in\mathbb{R}$. Let $K\subset M^n(c)$ be a $\lambda$-convex body, and let…

Conjecture on compact negatively curved space forms. Such a compact manifold exists if and only if $(p,q)$ occurs in the list

Let the ambient space be either the three-dimensional sphere or the three-dimensional hyperbolic space, and consider a standard double bubble enclosing two prescribed volumes. Cott…

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 $M^n_\kappa$ be the simply connected $n$-dimensional Riemannian manifold of constant sectional curvature $\kappa\in\{-1,1\}$, and let $\Omega\subset M^n_\kappa$ be a bounded co…

Let $n\geq 3$, let $\lambda\leq 0$, and let $K\in\mathcal{K}_0(\mathrm{Sp}^n(\lambda))$ be a convex body. Illumination-body convexity conjecture. For every $\delta>0$, the illumina…

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…

Space Form Conjecture. There exists a compact, complete pseudo-Riemannian manifold of signature $(p,q)$ with constant sectional curvature $1$ if and only if $(p,q)$ lies in the fol…

Let $M$ be an $n$-dimensional Riemannian manifold, let $c$ be a real number, and let $V_M(p,r)$ denote the volume of the geodesic ball of radius $r$ centered at $p$. Suppose that,…

Existence conjecture. The equation for $r$-harmonic curves admits solutions with non-constant geodesic curvature

Let $S^n$ be the standard sphere with $n\geq2$, and let $M^q$ be a space form of dimension $q$ with curvature at most zero. A Riemannian metric on a compact manifold with positive…

Let $l$ be a positive integer, let $\theta$ be the eigenvalue parameter, and let $a^{i_1i_3\cdots i_{2l-1}}_{i_2i_4\cdots i_{2l}}$ be a coefficient in the polynomial construction a…

Let $K$ be the curvature of a higher-dimensional space form, let $n$ be its dimension, and let $\theta$ denote the eigenvalue parameter appearing in the polynomial associated with…

Compact biconservative-surface conjecture. Any compact biconservative surface in $N^3(c)$ is $CMC$, that is, it has constant mean curvature.