As on arXiv
Abstract
We study the Ricci flow out of spaces with edge type conical singularities along a closed, embedded curve. Under the additional assumption that for each point of the curve, our space is locally modelled on the product of a fixed positively curved cone and a line, we show existence of a solution to Ricci flow for which converges back to the singular space as in the pointed Gromov-Hausdorff topology. We also prove curvature estimates for the solution and, for edge points, we show that the tangent flow at these points is a positively curved expanding Ricci soliton solution crossed with a line.
Comments: Final version, accepted at Communications in Analysis and Geometry
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities
Topics Singularities, Ricci solitons · Same author Lucas Lavoyer · Shared terms “conical singularities”, “modelled”
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Fixed geodesics of involutions on -spheres
Topics Limit spaces & RCD, Ricci solitons · Shared terms “positively curved”, “curved cone”, “curved expanding”
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Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program
Topics Weak & singular flows, Singularities, Ricci solitons · Same author Lucas Lavoyer · Shared terms “conical singularities”, “tangent flow”
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Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities
Topics Limit spaces & RCD, Singularities, Ricci solitons · Shared terms “conical singularities”
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On the rigidity of Ricci shrinkers
Topics Limit spaces & RCD, Singularities, Ricci solitons · Shared terms “Hausdorff topology”, “pointed Gromov”, “tangent flow”
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Mean Curvature Flow from Conical Singularities
Topics Weak & singular flows, Singularities · Shared terms “conical singularities”
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Closed mean curvature flows with asymptotically conical singularities
Topics Weak & singular flows, Singularities, Ricci solitons · Shared terms “conical singularities”
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Rigidity of the round cylinders in Ricci shrinkers
Topics Limit spaces & RCD, Ricci solitons · Shared terms “Hausdorff topology”, “pointed Gromov”, “Gromov Hausdorff”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.
- 6C.11Optimal Transport and Ricci Curvature
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.