15 problems

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 $(M,g)$ 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…

Let a generalized Wallach space (GWS) have parameters $a_1,a_2,a_3\in (0,1/2)$, and let $g$ be a metric of the form defined by with $\operatorname{Ric}>0$. Consider the normalized…

The $k$-core metric conjecture. These manifolds in fact admit $k$-core metrics with $k$ as given in these results.

Consider reduced spin graphs of rank $2$, together with their systems of invariants and the induced notion of $k$-equivalence. The preceding result shows that, for fixed nonzero in…

Different algebraic plumbing graphs can yield equivalent systems of invariants. To address this, a reduced form is introduced for the plumbing graphs, and the associated system of…

Let $(X,g,J)$ be a Kähler manifold with positive Ricci curvature, and consider the volume among Kähler metrics satisfying the relevant positive Ricci-curvature normalization. Folkl…

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 an ambient Riemannian manifold with positive Ricci curvature, and let $\Sigma\subset M$ be an index-$k$ embedded orientable compact minimal hypersurface. Linear Betti-nu…

Let $M$ be an ambient Riemannian manifold with positive Ricci curvature, and let $\Sigma\subset M$ be an index-one embedded orientable compact minimal hypersurface. Schoen's conjec…

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…

Let $M$ be a highly connected manifold of dimension $n>8$. The Stolz–Höhn conjecture. If $M$ admits a metric with positive Ricci curvature, then its Witten genus vanishes. The Witt…

Let $(M,g_0)$ be the compact three-dimensional Riemannian manifold with boundary considered in the Ricci-flow boundary value problem, and suppose that the boundary metric satisfies…