4 problems

Non-approximability conjecture. There exist Alexandrov spaces with curvature bounded below that admit no approximation by a sequence of Riemannian manifolds of fixed dimension with…

Let $(M,g)$ be an $n$-dimensional Riemannian manifold with Ricci curvature $Ric_M\ge (n-1)\kappa$. Yau's conjecture. There is a constant $c(n,\kappa)$ such that … This conjecture c…

Boundary curvature conjecture. It is still conjectured that $\partial X$ itself is an Alexandrov space with the same lower curvature bound $\kappa$ as $X$.

Anderson's conjecture. The results in dimension four should remain valid without the bounded topology and diameter assumptions.