Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912607675351040 |
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| author | Cao, Xiaodong Ribeiro Jr, Ernani Wondo, Hosea |
| author_facet | Cao, Xiaodong Ribeiro Jr, Ernani Wondo, Hosea |
| contents | We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient Shrinking Ricci Solitons and Modified Sectional Curvature Cao, Xiaodong Ribeiro Jr, Ernani Wondo, Hosea Differential Geometry We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds. |
| title | Gradient Shrinking Ricci Solitons and Modified Sectional Curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.20669 |