Curvature estimates for four-dimensional complete gradient expanding Ricci solitons
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911792853155840 |
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| author | Cao, Huai-Dong Liu, Tianbo |
| author_facet | Cao, Huai-Dong Liu, Tianbo |
| contents | In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set $K$). More precisely, we prove that the norm of the curvature tensor $Rm$ and its covariant derivative $\nabla Rm$ can be bounded by the scalar curvature $R$ by $|Rm|\le C_a R^a$ and $|\nabla Rm| \le C_a R^a$ (on $M\backslash K$), for any $0\le a <1$ and some constant $C_a >0$. Moreover, if the scalar curvature has at most polynomial decay at infinity, then $|Rm| \le C R$ (on $M\backslash K$). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton $(M^4, g, f)$ has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and $C^{1,α}$ asymptotic cones at infinity ($0 < α< 1$) according to Chen-Deruelle [20]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_09848 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Curvature estimates for four-dimensional complete gradient expanding Ricci solitons Cao, Huai-Dong Liu, Tianbo Differential Geometry 53C25, 53C21, 53E20 In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set $K$). More precisely, we prove that the norm of the curvature tensor $Rm$ and its covariant derivative $\nabla Rm$ can be bounded by the scalar curvature $R$ by $|Rm|\le C_a R^a$ and $|\nabla Rm| \le C_a R^a$ (on $M\backslash K$), for any $0\le a <1$ and some constant $C_a >0$. Moreover, if the scalar curvature has at most polynomial decay at infinity, then $|Rm| \le C R$ (on $M\backslash K$). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton $(M^4, g, f)$ has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and $C^{1,α}$ asymptotic cones at infinity ($0 < α< 1$) according to Chen-Deruelle [20]. |
| title | Curvature estimates for four-dimensional complete gradient expanding Ricci solitons |
| topic | Differential Geometry 53C25, 53C21, 53E20 |
| url | https://arxiv.org/abs/2111.09848 |