6 problems

Local model conjecture. The pair $(\mathcal H(M),\mathcal H_c(M))$ is locally homeomorphic to $(\square^\omega l_2,\boxdot^\omega l_2)$ at the identity $\mathrm{id}_M$ of $M$. In p…

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…

Rosenberg–Stolz conjecture. $Y\times\mathbb{R}^2$ does not admit a complete metric with uniformly positive scalar curvature.

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…

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