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 $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq 0$ for some $\alpha<\frac{4}{n-1}$ and mean-convex boundary, then it is either isometric to $\Sigma\times \mathbb{R}_{\geq 0}$ for a closed manifold $\Sigma$ with nonnegative Ricci curvature or it has no interior ends.
This paper proves a boundary splitting theorem under a spectral, rather than pointwise, lower bound for Ricci curvature.
Spectral splitting
Let $M^n$ have mean-convex boundary and $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq0$ for $\alpha<4/(n-1)$. If $M$ has an interior end, then it is isometric to $\Sigma\times\mathbb R_{\geq0}$ with $\Sigma$ closed and $\operatorname{Ric}_\Sigma\geq0$.
Dichotomy
The theorem allows a possibly noncompact boundary and gives a sharp alternative: either the product splitting occurs or every end of the manifold remains attached to the boundary.
Geometric application
In dimensions $n\leq4$, suitable nonnegative biRic curvature and mean convexity imply the same product-or-no-interior-end conclusion for complete two-sided stable free-boundary constant-mean-curvature immersions.
A positive solution of the spectral Ricci equation defines the weighted length functional $L_u^\alpha$. An interior end produces a free-boundary weighted minimizing ray, and a curve-capture argument constructs such a ray or line through every point. Its second variation forces the weight to be constant and hence upgrades spectral nonnegativity to pointwise $\operatorname{Ric}\geq0$; the classical boundary splitting theorem then finishes the proof.
My research interests include Geometric Analysis and Partial Differential Equations.