A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary

We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\geq 2$ has $λ_1(-αΔ+\operatorname{Ric})\geq 0$ for some $α<\frac{4}{n-1}$ and mean-convex boundary, then it is either isometric to $Σ\times \mathbb{R}_{\geq 0}$ for a closed manifold $Σ$ with nonnegative Ricci curvature or it has no interior ends.

💡 Research Summary

The paper establishes a Cheeger‑Gromoll‑type splitting theorem for non‑compact Riemannian manifolds that may have non‑compact boundary. The authors work under the hypothesis that the first eigenvalue of the operator (-\alpha\Delta+\operatorname{Ric}) is non‑negative, i.e. (\lambda_{1}(-\alpha\Delta+\operatorname{Ric})\ge0), for some constant (\alpha) satisfying (\alpha<4/(n-1)), where (n\ge2) is the dimension of the manifold. This condition is called “spectral non‑negative (\alpha)-Ricci curvature”. Together with the assumption that the boundary is mean‑convex (the second fundamental form with respect to the outward normal is non‑negative), the authors prove that the manifold (M) must fall into one of two mutually exclusive categories:

1. Product Splitting: (M) is isometric to a Riemannian product (\Sigma\times\mathbb R_{\ge0}), where (\Sigma) is a closed manifold whose Ricci curvature is non‑negative. In this case the boundary of (M) coincides with (\Sigma\times{0}) and is automatically compact.

2. No Interior Ends: (M) possesses no interior ends; all its ends are attached to the boundary. Consequently the boundary must be compact, even though compactness was not assumed a priori.

The proof proceeds through several novel steps. First, the spectral condition guarantees the existence of a positive function (u\in C^{2,\beta}(M)) solving \