
1. Diameter Pinching

1. Problem 1.1.

Does the Grove-Petersen-Wu Finiteness Theorem hold in dimension $4$, i.e. are there only finitely many diffeomorphism types in the class of $4$-dimensional Riemannian manifolds satisfying $$\sec_M \geq - \Lambda^2, \ \ \operatorname{diam}(M) \leq D, \ \ \operatorname{Vol}(M) \geq V ?$$
• Problem 1.2.

If $\sec_M \geq 1$ and $\operatorname{diam}(M) > \frac{\pi}{2}$, must $M$ be diffeomorphic to $S^n$?
• Problem 1.3.

Let $M^n$ be a manifold for which $\sec_M \geq 1$ and $\operatorname{diam} (M) \geq \frac{\pi}{2} - \varepsilon(n)$. Is $M$ homeomorphic to a manifold $M'$, where $\sec_{M'} \geq 1$, $\operatorname{diam}(M') \geq \frac{\pi}{2}$? This problem may be easier to address if we also assume that $\operatorname{Vol}(M) \geq V$ and $\varepsilon = \varepsilon(n, V)$.

Cite this as: AimPL: Manifolds with non-negative sectional curvature, available at http://aimpl.org/nnsectcurvature.