6 problems

Let $(M,g)$ satisfy the hypotheses of the preceding one-endedness conjecture, and suppose that its Ricci curvature agrees with that of a paraboloid for $r\leq r_0$. Paraboloid rigi…

Let $(M,g)$ be a complete, noncompact manifold with finitely many ends, each end outside a compact set diffeomorphic to a spherical shell. Assume that $M$ has everywhere positive R…

Loop-space conjecture. The space $\operatorname{\mathcal{R}^\text{Cores}}(\operatorname{S}^n)$ equipped with $\mu$ is an associative $H$-space and is homotopy equivalent to a loop…

Let $M$ be a compact three-dimensional manifold with positive Ricci curvature. Suppose that $S_1$ and $S_2$ are embedded constant mean curvature (CMC) spheres in $M$ that kiss at a…

Let $M$ be a three-dimensional Riemannian manifold with positive Ricci curvature, and let $S$ be a self-touching constant mean curvature (CMC) surface in $M$. Finiteness conjecture…

Let $M$ be a highly connected manifold admitting a positive scalar curvature metric and with vanishing Witten genus. A homotopy-sphere Ricci-positivity conjecture. There exists a h…