Cheng's eigenvalue comparison on metric measure spaces and applications
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916880619405312 |
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| author | De Luca, G. Bruno De Ponti, Nicolò Mondino, Andrea Tomasiello, Alessandro |
| author_facet | De Luca, G. Bruno De Ponti, Nicolò Mondino, Andrea Tomasiello, Alessandro |
| contents | Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth setting of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense, via optimal transport. Rigidity and stability statements are provided for $\mathsf{RCD}^{\star}(K,N)$ spaces; the stability seems to be new even for smooth Riemannian manifolds. We then present some mathematical and physical applications: in the former, we obtain an upper bound on the $j^{th}$ Neumann eigenvalue in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces and a bound on the essential spectrum in non-compact $\mathsf{RCD}^{\star}(K,N)$ spaces; in the latter, the eigenvalue bounds correspond to general upper bounds on the masses of the spin-2 Kaluza-Klein excitations around general warped compactifications of higher-dimensional theories of gravity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cheng's eigenvalue comparison on metric measure spaces and applications De Luca, G. Bruno De Ponti, Nicolò Mondino, Andrea Tomasiello, Alessandro Spectral Theory High Energy Physics - Theory Differential Geometry Metric Geometry Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth setting of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense, via optimal transport. Rigidity and stability statements are provided for $\mathsf{RCD}^{\star}(K,N)$ spaces; the stability seems to be new even for smooth Riemannian manifolds. We then present some mathematical and physical applications: in the former, we obtain an upper bound on the $j^{th}$ Neumann eigenvalue in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces and a bound on the essential spectrum in non-compact $\mathsf{RCD}^{\star}(K,N)$ spaces; in the latter, the eigenvalue bounds correspond to general upper bounds on the masses of the spin-2 Kaluza-Klein excitations around general warped compactifications of higher-dimensional theories of gravity. |
| title | Cheng's eigenvalue comparison on metric measure spaces and applications |
| topic | Spectral Theory High Energy Physics - Theory Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2507.23671 |