Convergence of Ricci flow and long-time existence of Harmonic map heat flow
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| Format: | Preprint |
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2025
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| _version_ | 1866910902795632640 |
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| author | Choi, Kyeongsu Lai, Yi |
| author_facet | Choi, Kyeongsu Lai, Yi |
| contents | For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms.
We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round $\mathbb S^n$ singularity converges at least at the rate $(-t)^{\frac{n+1}{n-1}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_02804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of Ricci flow and long-time existence of Harmonic map heat flow Choi, Kyeongsu Lai, Yi Differential Geometry Analysis of PDEs 53E20 For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms. We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round $\mathbb S^n$ singularity converges at least at the rate $(-t)^{\frac{n+1}{n-1}}$. |
| title | Convergence of Ricci flow and long-time existence of Harmonic map heat flow |
| topic | Differential Geometry Analysis of PDEs 53E20 |
| url | https://arxiv.org/abs/2504.02804 |