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A steady length function for Ricci flows. (English) Zbl 07301345
Summary: A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons. A similar quantity was found by Feldman, Ilmanen, and Ni [J. Geom. Anal. 15 (2005), pp. 49-62] which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification of the steady soliton equation.
MSC:
53E20 Ricci flows
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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[1] Feldman, Michael; Ilmanen, Tom; Ni, Lei, Entropy and reduced distance for Ricci expanders, J. Geom. Anal., 15, 1, 49-62 (2005) · Zbl 1071.53040
[2] Huisken, Gerhard, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom., 31, 1, 285-299 (1990) · Zbl 0694.53005
[3] Kleiner, Bruce; Lott, John, Notes on Perelman’s papers, Geom. Topol., 12, 5, 2587-2855 (2008) · Zbl 1204.53033
[4] Perelman G. Perelman, \newblock The entropy formula for the Ricci flow and its geometric applications, \newblock ArXiv e-prints, (Feb. 2008). \arXiv math/0211159 · Zbl 1130.53001
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