Einstein-type structures, Besse's conjecture and a uniqueness result for a $φ$-CPE metric in its conformal class
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| Format: | Preprint |
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2022
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| _version_ | 1866914640502456320 |
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| author | Colombo, Giulio Mari, Luciano Rigoli, Marco |
| author_facet | Colombo, Giulio Mari, Luciano Rigoli, Marco |
| contents | In this paper, we study an extension of the CPE conjecture to manifolds $M$ which support a structure relating curvature to the geometry of a smooth map $φ: M \to N$. The resulting system, denoted by $(φ-\mathrm{CPE})$, is natural from the variational viewpoint and describes stationary points for the integrated $φ$-scalar curvature functional restricted to metrics with unit volume and constant $φ$-scalar curvature. We prove both a rigidity statement for solutions to $(φ-\mathrm{CPE})$ in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting $(φ-\mathrm{CPE})$ with $φ$ a harmonic map. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_00263 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Einstein-type structures, Besse's conjecture and a uniqueness result for a $φ$-CPE metric in its conformal class Colombo, Giulio Mari, Luciano Rigoli, Marco Differential Geometry Analysis of PDEs In this paper, we study an extension of the CPE conjecture to manifolds $M$ which support a structure relating curvature to the geometry of a smooth map $φ: M \to N$. The resulting system, denoted by $(φ-\mathrm{CPE})$, is natural from the variational viewpoint and describes stationary points for the integrated $φ$-scalar curvature functional restricted to metrics with unit volume and constant $φ$-scalar curvature. We prove both a rigidity statement for solutions to $(φ-\mathrm{CPE})$ in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting $(φ-\mathrm{CPE})$ with $φ$ a harmonic map. |
| title | Einstein-type structures, Besse's conjecture and a uniqueness result for a $φ$-CPE metric in its conformal class |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2201.00263 |