Orbifold singularity formation along ancient and immortal Ricci flows
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| Format: | Preprint |
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2024
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| _version_ | 1866910793812934656 |
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| author | Deruelle, Alix Ozuch, Tristan |
| author_facet | Deruelle, Alix Ozuch, Tristan |
| contents | In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension $4$ with Einstein orbifolds as tangent flows at infinity. For instance, for any $k\in\mathbb{N}_0$, we obtain continuous families of non-isometric ancient Ricci flows on $\#k(\mathbb{S}^2\times \mathbb{S}^2)$ depending on a number of parameters growing linearly in $k$, and a family of half-PIC ancient Ricci flows on $\mathbb{CP}^2\#\mathbb{CP}^2$.
The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orbifold singularity formation along ancient and immortal Ricci flows Deruelle, Alix Ozuch, Tristan Differential Geometry Analysis of PDEs In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension $4$ with Einstein orbifolds as tangent flows at infinity. For instance, for any $k\in\mathbb{N}_0$, we obtain continuous families of non-isometric ancient Ricci flows on $\#k(\mathbb{S}^2\times \mathbb{S}^2)$ depending on a number of parameters growing linearly in $k$, and a family of half-PIC ancient Ricci flows on $\mathbb{CP}^2\#\mathbb{CP}^2$. The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics. |
| title | Orbifold singularity formation along ancient and immortal Ricci flows |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2410.16075 |