Pyramid Ricci flow in higher dimensions
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Mathematische Zeitschrift
Pyramid Ricci flow in higher dimensions Andrew D. McLeod1 · Peter M. Topping2 Received: 24 August 2019 / Accepted: 21 October 2019 © The Author(s) 2020
Abstract In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold (M n , g0 ) that is PIC1, or more generally satisfies a lower curvature bound KIC1 ≥ −α0 . That is, instead of constructing a flow on M × [0, T ], we construct it on a subset of space-time that is a union of parabolic cylinders Bg0 (x0 , k) × [0, Tk ] for each k ∈ N, where Tk ↓ 0, and prove estimates on the curvature and Riemannian distance. More generally, we construct a pyramid Ricci flow starting with any noncollapsed IC1 -limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder.
1 Introduction A central issue in differential geometry is to understand Riemannian manifolds with lower curvature bounds. One of the important tasks in this direction is to understand the topological implications of such geometric bounds. Another, which is the main focus of this paper, is to understand the structure of Gromov–Hausdorff limits of sequences of manifolds satisfying a uniform lower curvature bound. There is some choice as to the precise notion of curvature bound to consider. Imposing a uniform lower bound on the sectional curvatures gives limits that are Alexandrov spaces, studied since the middle of the twentieth century, and about which we now have a great deal of information, e.g. [5]. In practice, we often know a uniform lower bound not for each sectional curvature, but for a suitable average of sectional curvatures, and the instance that has received the most attention is the case of limits of manifolds with a uniform lower Ricci bound. Such Ricci limit spaces have been studied extensively since the work of CheegerColding, starting in the 1990s, and have been widely applied, for example in the study of Einstein manifolds, [7]. One result that is particularly relevant to the present paper is the
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Peter M. Topping [email protected] https://homepages.warwick.ac.uk/∼maseq/ Andrew D. McLeod [email protected] https://iris.ucl.ac.uk/iris/browse/profile?upi=AMCLE50
1
Department of Mathematics, University College London, London WC1H 0AY, UK
2
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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A. D. McLeod, P. M. Topping
topological regularity of (non-collapsed) three-dimensional Ricci limit spaces in the sense that they are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder [11,15,22]. This paper is concerned principally with a way of averaging sectional curvatures that is less familiar than Ricci curvature. Positivity of this average is generally referred to as PIC1, with this concept first appearing in the seminal work of Micallef and Moore [16] that was principally concerned with the weaker notion of positive isotropic curvature, itself now abbreviated as PIC. We will give the
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