8 problems

A pseudo-Riemannian manifold of signature $(p,q)$ is understood here to be compact and complete, with constant sectional curvature $1$. Space form conjecture. There exists such a m…

Polynomiality conjecture. Each coefficient $a^{i_1i_3\cdots i_{2l-1}}_{i_2i_4\cdots i_{2l}}$ is a polynomial in $\theta$ of degree $l$. Conjecturally, the constant in the displayed…

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 $n\geqslant 2$, 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…

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 a pseudo-Riemannian space form of signature $(p,q)$ be a pseudo-Riemannian manifold with constant sectional curvature $\kappa$, where $p,q\neq 0$. Kobayashi's pseudo-Riemannian…