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

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.

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

Thu, 16 Jan 2025 04:51:14 +0800

Main results

This work proves scalar-curvature splitting theorems for noncompact three-manifolds with mean-convex boundary, using noncompact minimizing free-boundary and capillary surfaces as the rigidifying objects.

Free-boundary splitting

A minimizing half-cylinder or strip forces flatness

If the manifold contains an absolutely area-minimizing free-boundary half-cylinder or strip, then it is flat. Up to scaling, the resulting models are covered by $\mathbb S^1\times\mathbb R^2_+$ or are isometric to $[0,1]\times\mathbb R^2$.

Capillary extension

The theorem holds for every contact angle

For $\theta\in(0,\pi)$, an absolutely $\theta$-energy-minimizing half-cylinder gives the corresponding tilted flat model; the strip statement also holds when its two boundary curves lie on different boundary components.

No compact-boundary hypothesis

The ambient boundary may be noncompact or disconnected

The argument requires neither compactness nor a prescribed number of components of $\partial M$, extending earlier boundary splitting theorems beyond their usual global assumptions.

Proof strategy

Plateau-type problems in carefully chosen prism-shaped domains produce a sequence of minimizing capillary surfaces near the given noncompact surface. Stability classifies the possible limits, while energy comparisons rule out unwanted disk, half-plane, plane, and cylinder degenerations. The surviving half-cylinders or strips form a rigid family; equality in the scalar-curvature and boundary inequalities then propagates to an ambient flat product.

Splitting | Gaoming Wang

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

Main results

An interior end forces a half-product

Otherwise there are no interior ends

Stable free-boundary CMC hypersurfaces inherit the dichotomy

Proof strategy

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

Main results

A minimizing half-cylinder or strip forces flatness

The theorem holds for every contact angle

The ambient boundary may be noncompact or disconnected

Proof strategy