Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909976102961152 |
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| author | Bednarek, Alexander |
| author_facet | Bednarek, Alexander |
| contents | We consider the Kähler-Ricci flow $(X, ω(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the Kähler manifold $X$ to some analytic variety $Y$ which admits a Kähler metric on a neighbourhood of the image of $X$ and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., $[ω_0] \in H^{1,1}(X,\mathbb{Q})$. Under these assumptions we prove an $L^4$-like estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type $I$ in the $L^2$-sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11608 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities Bednarek, Alexander Differential Geometry Complex Variables 53E30, 32W50 We consider the Kähler-Ricci flow $(X, ω(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the Kähler manifold $X$ to some analytic variety $Y$ which admits a Kähler metric on a neighbourhood of the image of $X$ and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., $[ω_0] \in H^{1,1}(X,\mathbb{Q})$. Under these assumptions we prove an $L^4$-like estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type $I$ in the $L^2$-sense. |
| title | Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities |
| topic | Differential Geometry Complex Variables 53E30, 32W50 |
| url | https://arxiv.org/abs/2409.11608 |