Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908948420886528 |
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| author | Catino, Giovanni Mari, Luciano Mastrolia, Paolo Roncoroni, Alberto |
| author_facet | Catino, Giovanni Mari, Luciano Mastrolia, Paolo Roncoroni, Alberto |
| contents | In this paper we prove general criticality criteria for operators $Δ+ V$ on manifolds with more than one end, where $V$ bounds the Ricci curvature, and a related spectral splitting theorem extending Cheeger-Gromoll's one. Our results give new insight on Li-Wang's theory of manifolds with a weighted Poincaré inequality. We apply them to study stable and $δ$-stable minimal hypersurfaces in manifolds with non-negative bi-Ricci or sectional curvature, in ambient dimension up to $5$ and $6$, respectively. In the special case where the ambient space is $\mathbb{R}^4$, we prove that a $1/3$-stable minimal hypersurface must either have one end or be a catenoid, and that proper, $δ$-stable minimal hypersurfaces with $δ> 1/3$ must be hyperplanes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12631 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces Catino, Giovanni Mari, Luciano Mastrolia, Paolo Roncoroni, Alberto Differential Geometry Analysis of PDEs In this paper we prove general criticality criteria for operators $Δ+ V$ on manifolds with more than one end, where $V$ bounds the Ricci curvature, and a related spectral splitting theorem extending Cheeger-Gromoll's one. Our results give new insight on Li-Wang's theory of manifolds with a weighted Poincaré inequality. We apply them to study stable and $δ$-stable minimal hypersurfaces in manifolds with non-negative bi-Ricci or sectional curvature, in ambient dimension up to $5$ and $6$, respectively. In the special case where the ambient space is $\mathbb{R}^4$, we prove that a $1/3$-stable minimal hypersurface must either have one end or be a catenoid, and that proper, $δ$-stable minimal hypersurfaces with $δ> 1/3$ must be hyperplanes. |
| title | Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2412.12631 |