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

Fri, 23 Jan 2026 11:51:14 +0800

Main results

This paper proves a boundary splitting theorem under a spectral, rather than pointwise, lower bound for Ricci curvature.

Spectral splitting

An interior end forces a half-product

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

Otherwise there are no interior ends

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

Stable free-boundary CMC hypersurfaces inherit the dichotomy

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.

Proof strategy