10 problems

Let $(M^{n},g)$ be a closed, simply connected Riemannian manifold of dimension $n\ge 4$. Define the minimum sectional curvature and normalized scalar curvature at $x\in M$ by … Her…

Let $(M^n,g,f)$ be a CPE metric, meaning that $(M^n,g)$ is a closed, oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature and $f:M^n\to\mathbb{R}$ is…

Let $M^n$ be a closed $n$-dimensional Riemannian manifold, and let $n+\frac{n-2}{n}$ be the positivity threshold for its curvature operator of the second kind. Li's conjecture. If…

Lei–Xu's conjecture. If

Xu–Gu's conjecture. If

Gu–Xu's conjecture. If

Let $F_{0}:M^{n}\to\mathbb{S}^{n+q}$ be an $n$-dimensional closed submanifold satisfying … where $h$ is the second fundamental form. Lawson–Simons conjecture. The mean curvature fl…

Let $M_{0}$ be an $n$-dimensional complete submanifold in the sphere $\mathbb{S}^{n+q}\left(1/\sqrt{c}\right)$. Let $h$ be its second fundamental form, $H$ its mean curvature vecto…

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…

An exotic sphere is a smooth manifold homeomorphic but not diffeomorphic to a standard sphere. A Riemannian metric is strictly $1/4$-pinched in the global sense when its sectional…