15 problems

A compact, simply connected Riemannian manifold $M$ is $1/4$-pinched when its sectional curvatures satisfy the quarter-pinching condition. A minimal submanifold is stable when the…

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 $M$ be a compact simply connected differentiable manifold, and fix $\delta\in(0,1]$. Consider all $\delta$-pinched Riemannian structures on $M$, and let $i(M,g)$ denote the inj…

Second pinching conjecture. If $S>n$, then $S\geq 2n$.

Florio–Hryniewicz conjecture. The good bound for $\delta$ should be $1/4$: among arbitrary positively curved $2$-spheres, the geodesic flow on the unit tangent bundle should be lef…

Let $(M,g,f)$ be an $n$-dimensional gradient shrinking Ricci soliton, or shrinker, satisfying … The shrinker is noncollapsed if it satisfies the noncollapsing condition, and it has…

Let $M$ be a surface of codimension two in $\mathbb{R}^{4}$ satisfying the pinching condition … The Baker–Nguyen conjecture. The Clifford torus is the true obstruction to the conve…

Luo–Sun pinching conjecture. Then $M$ is either a totally geodesic sphere or the Calabi torus.

Gu–Xu's conjecture. If

Yau's conjecture. If

Let $M^n$, with $n\geq4$, be an even-dimensional compact and simply connected Riemannian manifold. Let $K_M$ denote the curvature quantity used in the source, let $R_0$ be the norm…

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. Let $K_M$ denote the relevant curvature quantity and let $R_0$ be the normalized scalar curvature. Yau's second con…

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…