As on arXiv
Abstract
We study the uniqueness problem for the Kähler-Ricci flow with a conical initial condition. Given a complete gradient expanding Kähler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we establish that any complete solution to the Kahler-Ricci flow emerging from the soliton's tangent cone at infinity–appearing as a Kähler cone–must coincide with the forward self-similar Kähler-Ricci flow associated with the soliton, provided certain conditions hold. Specifically, if its Kähler form remains in the same cohomology class as that of the soliton's self-similar Kähler-Ricci flow, its full Riemann curvature operator is bounded for each fixed positive time, its Ricci curvature is bounded from above by A/t, its scalar curvature is bounded from below by -A/t, and it shares a same Killing vector field with the soliton metric. This paper gives a partial answer to a question in paper of Feldman-Ilmanen-Knopf, and generalizes the earlier work of Conlon-Deruelle and the work of Conlon-Deruelle-Sun.
Comments: 30 pages, all comments are welcome!
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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Stability of asymptotically conical gradient Kähler-Ricci expanders
Topics Kähler–Ricci solitons, Kähler–Ricci flow, Boundaries & non-compact · Same author Longteng Chen · Shared terms “asymptotically conical”, “expanding Kähler”, “gradient expanding”
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Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow
Topics Kähler–Ricci flow, Boundaries & non-compact, Kähler geometry · Same author Longteng Chen · Shared terms “asymptotically conical”
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On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces
Topics Kähler–Ricci solitons, Kähler–Ricci flow, Scalar curvature · Shared terms “Feldman Ilmanen”, “Ilmanen Knopf”, “vector field”
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A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity
Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “tangent cone”, “Kähler cone”, “asymptotically conical”
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Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities
Topics Kähler–Ricci flow, Kähler geometry, Ricci solitons · Same author Longteng Chen · Shared terms “flow emerging”, “Kähler cone”
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Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones
Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “Kähler cone”, “asymptotically conical”
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Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program
Topics Kähler–Ricci flow, Boundaries & non-compact, Kähler geometry · Same author Longteng Chen · Shared terms “conical Kähler”, “asymptotically conical”
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Linear Stability of Steady and Expanding Kähler-Ricci Solitons
Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “expanding Kähler”, “asymptotically conical”
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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
- 5A.6Where This Track Leads
The Kähler–Ricci flow, and when to come back to it.
- 6C.4The Yamabe Problem and Critical Exponents
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
- 9B.6Scalar Curvature and Topology
Positive scalar curvature, its obstructions, and the positive mass theorem.