Pyramid Ricci flow in higher dimensions

McLeod, Andrew; Topping, Peter M.

doi:10.1007/s00209-020-02472-1

Cited by 13 publications

(5 citation statements)

References 16 publications

(27 reference statements)

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“…Added in proof: Between acceptance and final publication of this paper, generalisations of our results to higher dimensions have been published in [MT20].…”

Section: Appendix a Results From Simon-topping Papersmentioning

confidence: 92%

Global regularity of three-dimensional Ricci limit spaces

McLeod¹,

Topping²

2022

Trans. Amer. Math. Soc. Ser. B

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Miles Simon and the second author, in their recent work [Geom. Topol. 25 (2021), pp. 913–948], established a local bi-Hölder correspondence between weakly noncollapsed Ricci limit spaces in three dimensions and smooth manifolds. In particular, any open ball of finite radius in such a limit space must be bi-Hölder homeomorphic to some open subset of a complete smooth Riemannian three-manifold. In this work we build on the technology from Simon and the second author in [J. Differential Geometry, to appear] and [Geom. Topol. 25, (2021), pp. 913–948] to improve this local correspondence to a global-local correspondence. That is, we construct a smooth three-manifold M , M , and prove that the entire (weakly) noncollapsed three-dimensional Ricci limit space is homeomorphic to M M via a globally-defined homeomorphism that is bi-Hölder once restricted to any compact subset. Here the bi-Hölder regularity is with respect to the distance d g d_g on M , M, where g g is any smooth complete metric on M M . A key step in our proof is the construction of local pyramid Ricci flows, existing on uniform regions of spacetime, that are inspired by Hochard’s partial Ricci flows in the paper by Raphaël Hochard [Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, https://arxiv.org/abs/1603.08726, 2016]. Suppose ( M , g 0 , x 0 ) \left ( M , g_0 , x_0 \right ) is a complete smooth pointed Riemannian three-manifold that is (weakly) noncollapsed and satisfies a lower Ricci bound. Then, given any k ∈ N , k \in \mathbb {N}, we construct a smooth Ricci flow g ( t ) g(t) living on a subset of spacetime that contains, for each j ∈ { 1 , … , k } j \in \left \{1 , \ldots , k \right \} , a cylinder B g 0 ( x 0 , j ) × [ 0 , T j ] \mathbb {B}_{g_0} \left ( x_0 , j \right )\times [0,T_j] , where T j T_j is dependent only on the Ricci lower bound, the (weakly) noncollapsed volume lower bound and the radius j j (in particular independent of k k ) and with the property that g ( 0 ) = g 0 g(0)=g_0 throughout B g 0 ( x 0 , k ) \mathbb {B}_{g_0} \left ( x_0 , k \right ) .

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“…Added in proof: Between acceptance and final publication of this paper, generalisations of our results to higher dimensions have been published in [MT20].…”

Section: Appendix a Results From Simon-topping Papersmentioning

confidence: 92%

Global regularity of three-dimensional Ricci limit spaces

McLeod¹,

Topping²

2022

Trans. Amer. Math. Soc. Ser. B

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“…The existence of such a flow was proved in three dimensions in [23], and could also be derived from a combination of [11] and [22]. The proof was extended by Lai [14] and Hochard [12] to the analogous result in higher dimensions, and the following version of their extensions is a special case of [18,Theorem 3.3 ].…”

Section: Time Zero Regularity In the Presence Of An Ic 1 Lower Boundmentioning

confidence: 98%

Time zero regularity of Ricci flow

Lee¹,

Topping²

2022

Preprint

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We consider the problem of when a smooth Ricci flow, for positive time, that attains smooth initial data in a weak sense must be smooth down to the initial time. We develop an example where this fails. We prove a positive result in the case that the flow satisfies a lower KIC 1 curvature bound, equivalent to a lower Ricci bound in three dimensions. As an application, we prove that Gromov-Hausdorff limits of WPIC1 manifolds are WPIC1.

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“…Combining the maximum principle with the partial Ricci flow method [12,26], Lai [14] constructed a complete Ricci flow solution starting from a complete noncollapsed metric which is of almost weakly PIC 1 , and remains almost weakly PIC 1 for a short time. In [21], McLeod and Topping combined Theorem 1.2, Lai's Ricci flow solutions [14] and the techniques developed in their earlier work [20] to obtain a smooth structure on the noncollapsed IC 1 -limit space. In [16], the authors used Theorem 1.2 to construct a local Kähler-Ricci flow starting from a noncollapsed Kähler manifold with almost nonnegative curvature and improve a result of Liu [18] on the complex structure of the corresponding Gromov-Hausdorff limit of this class of Kähler manifolds.…”

Section: Theorem 12 Let G(t) Be a Smooth Ricci Flow On M N × [0 T] Wh...mentioning

confidence: 99%

Some local maximum principles along Ricci flows

Lee¹,

Tam

2020

Can. J. Math.-J. Can. Math.

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In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ for some constant ${\alpha }>0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ .

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Pyramid Ricci flow in higher dimensions

Cited by 13 publications

References 16 publications

Global regularity of three-dimensional Ricci limit spaces

Global regularity of three-dimensional Ricci limit spaces

Time zero regularity of Ricci flow

Some local maximum principles along Ricci flows

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