19 problems

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…

Nishikawa's conjecture. If $M$ has positive curvature operator of the second kind, then $M$ is diffeomorphic to a spherical space form; if $M$ has nonnegative curvature operator of…

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 $M_{0}$ 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…

Ricci sphere-threshold conjecture. The result remains true for

Let $\mathcal{F}$ be a $q$-codimensional complete Riemannian foliation of a connected manifold $M$, with transverse sectional curvature satisfying $1/4<\sec_{\mathcal{F}}\leq 1$ an…

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

Conjecture 5.7. The pointwise bound on the second fundamental form should force either finite-time convergence to a round point or infinite-time convergence to a totally geodesic s…

Let $(M,g_0)$ be an $n$-dimensional compact submanifold of an $(n+p)$-dimensional space form $F^{n+p}(c)$, where $n\geq5$ and $c+H^2>0$. Let $\operatorname{Ric}_{M}$ denote the Ric…

Let $M^n$ be an odd-dimensional compact minimal submanifold in $S^{n+p}$. Write $UM=\bigcup_{x\in M}U_xM$, where $U_xM=\{u\in T_xM:\|u\|=1\}$, and let $h$ be the second fundamental…

Xu–Gu's conjecture. If

Gu–Xu's conjecture. If

Yau's conjecture. If

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…

Let $M^n$, with $n\geq4$, be a compact Riemannian manifold. Let $R_0$ be its normalized scalar curvature and $K_{\max}$ its maximum sectional curvature. Scalar-curvature pinching c…

Let $M^n$ be a compact and simply connected Riemannian manifold, and let $R_0$ denote its normalized scalar curvature. Yau's first conjecture. If … then $M$ is diffeomorphic to…

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…

Let $(M,g)$ be a compact, simply connected Riemannian manifold whose sectional curvature satisfies $\delta \leq K \leq 1$ for a value of $\delta$ close to $1$. Hopf's pinching conj…