Capillary prisms, Coxeter gluing, and the $\pi_2$-systole of 3-manifolds with positive scalar curvature

Abstract

We study the $\pi_2$-systole of positive scalar curvature $3$-manifolds via a local-to-global approach. The tessellated local building blocks are capillary prisms $M$: Riemannian cylinders enclosed by mean convex surfaces meeting at prescribed dihedral angles. For a class of such prisms, called energy-essential prisms, we obtain angle-sensitive relative $\pi_2$-systole estimates. In particular, if the upper capillary angle is at most $\alpha \in (0,\tfrac{\pi}{2}]$, then \[\mathrm{sys}_2(M,\partial_0 M, g)\cdot \inf R_g \lesssim |\log \alpha|^{-1},\qquad \alpha\to 0.\] The proof relies on the monotonicity of a new quasi-local mass. The local-to-global step is achieved via Coxeter gluing of copies of capillary prisms. We prove a general Coxeter gluing and smoothing theorem for scalar curvature: under Coxeter compatibility of the corner strata and nonnegative mean curvature jump conditions along the glued facets, we prove that the resulting piecewise smooth manifold can be smoothened while preserving the scalar curvature lower bound. This theorem gives, to our knowledge, the first general rigorous formulation and proof of the Coxeter-polyhedral smoothing principle for scalar curvature lower bounds, a principle that has long been used heuristically in positive scalar curvature geometry. As an application, we construct smooth positive scalar curvature metrics with large $\pi_2$-systole on connected sums of copies of $S^2\times S^1$ and lens spaces that are not covered by $S^2\times R$. These examples give the first explicit metrics with quantitatively large $\pi_2$-systole on these topologies. We also discuss rigidity and non-rigidity phenomena for the local $\pi_2$-systolic estimates.