A splitting theorem for 3-manifold with nonnegative scalar curvature and mean-convex boundary

TL;DR

The paper proves a splitting-type rigidity theorem for orientable, complete 3-manifolds with nonnegative scalar curvature and mean-convex boundary, under the presence of an absolutely area-minimizing half-cylinder or strip (or θ-energy-minimizing variants) in the manifold. The approach centers on constructing and analyzing stable capillary minimal surfaces via a variational framework for θ-energy, including limits of Plateau-type problems in suitably chosen domains, to produce limiting surfaces that enforce flatness and a product-like geometry. Key results show that, up to scaling and finite covers, the manifold must be isometric to a wedge product $S^1\times\mathbb{R}^2_+$ or to $S^1\times[0,1]\times\mathbb{R}$, with the former corresponding to half-cylindrical splittings and the latter to strip-type splittings, and they include θ-dependent refinements and topological considerations. These findings extend classical splitting theorems to manifolds with boundary under capillary-energy constraints and connect to stability and rigidity phenomena of stable minimal surfaces in scalar-curvature-vanishing regions.

Abstract

We show that a Riemannian 3-manifold with nonnegative scalar curvature and mean-convex boundary is flat if it contains an absolutely area-minimizing (in the free boundary sense) half-cylinder or strip. Analogous results also hold for a $θ$-energy-minimizing half-cylinder, or, under certain topological assumptions, a $θ$-energy-minimizing strip for $θ\in (0,π)$.

Figures (8)

• Figure 1: A locally wedge domain
• Figure 2: The choice of $U_h^\delta$ and $B_h^\delta$ when $S$ is a strip
• Figure 3: The choice of $U_h^\delta$ and $B_h^\delta$ when $S$ is a half-cylinder
• Figure 4: The shape of $\Sigma(r,t,h,\delta)$ in the case of strip
• Figure 5: A disc converging to a half-cylinder

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof : Proof of Lemma \ref{['lemma:flatAnnulus']}
• Lemma 2.6
• Definition 3.1