Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary
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2026
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| author | Antonelli, Gioacchino Li, Yangyang Sweeney Jr, Paul |
| author_facet | Antonelli, Gioacchino Li, Yangyang Sweeney Jr, Paul |
| contents | We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the conditions \[ λ_1(-γΔ+\mathrm{Ric})\geq 0,\qquad H_{\partial M}\geq 0, \] and in the sharp range $0\leq γ<4$ if $n=2$, and $0\leqγ<\frac{n-1}{n-2}$ if $n\geq3$, a (possibly noncompact) complete manifold with disconnected boundary, with at least one compact boundary component, must split isometrically as a product $[0,L]\times Σ$.
Our second main contribution is a topological rigidity result for the relative fundamental group $π_1(M,\partial M)$, combined with a deep theorem of Lawson--Michelsohn. We prove that, in dimensions $n\neq4$, any compact manifold with boundary satisfying the two inequalities above, with at least one of them strict, admits a metric with positive sectional curvature and strictly mean-convex boundary, provided $γ\geq0$ if $n=2$, and $0\leqγ\leq\frac{n-1}{n-2}$ if $n\geq3$. This range of $γ$ is sharp for the latter result to hold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11384 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary Antonelli, Gioacchino Li, Yangyang Sweeney Jr, Paul Differential Geometry Analysis of PDEs We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the conditions \[ λ_1(-γΔ+\mathrm{Ric})\geq 0,\qquad H_{\partial M}\geq 0, \] and in the sharp range $0\leq γ<4$ if $n=2$, and $0\leqγ<\frac{n-1}{n-2}$ if $n\geq3$, a (possibly noncompact) complete manifold with disconnected boundary, with at least one compact boundary component, must split isometrically as a product $[0,L]\times Σ$. Our second main contribution is a topological rigidity result for the relative fundamental group $π_1(M,\partial M)$, combined with a deep theorem of Lawson--Michelsohn. We prove that, in dimensions $n\neq4$, any compact manifold with boundary satisfying the two inequalities above, with at least one of them strict, admits a metric with positive sectional curvature and strictly mean-convex boundary, provided $γ\geq0$ if $n=2$, and $0\leqγ\leq\frac{n-1}{n-2}$ if $n\geq3$. This range of $γ$ is sharp for the latter result to hold. |
| title | Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2605.11384 |