6 problems

Let $(M^n,g_0)$ be a complete WPIC1 manifold with $n\geq 3$. WPIC1 Ricci-flow existence conjecture. There exists a complete WPIC1 Ricci flow $g(t)$ on $M$ for $t\in[0,T)$, for some…

Let $\overline{E}$ be a holomorphic vector bundle on $\overline{X}$ such that $\overline{E}|_D$ is $c_1(N_D)$-polystable and $\overline{E}$ is $(c_1(D),[\omega_0])$-stable. Let…

Let $M$ be a complete non-compact Riemannian manifold with locally convex ends, meaning that each end can be cut off by a bounded hypersurface that is locally convex toward infinit…

Gromov's compactness conjecture. This is a $\sideset{}{^1}\varprojlim$ phenomenon and naturally leads one to conjecture that Gromov's compactness question has a positive answer pro…

Let $M$ be a non-compact manifold with boundary, and let $\mathcal{A}_0$ and $\mathcal{A}_1$ be the boundary operators for the relevant strongly Callias-type operators. Suppose tha…

Signature-index invariance conjecture. The large-scale indices should coincide: