As on arXiv
Abstract
We study a Boltzmann's type entropy functional (which appeared in existing literature) defined on Kähler metrics of a fixed Kähler class. The critical points of this functional are gradient Kähler-Ricci solitons, and the functional was known to be monotonically increasing along the Kähler-Ricci flow in the canonical class. In this article, we derive and analyze the second variation formula for this entropy functional, and show that all gradient Kähler-Ricci solitons are stable with respect to this entropy functional. Furthermore, using this result, we give a new proof that gradient shrinking Kähler-Ricci solitons are stable with respect to the Perelman's entropy in a fixed Kähler class.
- Submitted
- May 25, 2016
- Subjects
- math.DG
Comments: 15 pages; 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 2016, plus the canon.
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A Spinorial Perelman's Functional: Critical Points and Gradient Flow
Topics Perelman entropy, Ricci solitons · Same author Frederick Tsz-Ho Fong · Shared terms “critical points”, “variation formula”
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Kähler-Ricci flow for deformed complex structures
Topics Perelman entropy, Kähler–Ricci solitons, Kähler–Ricci flow · Shared terms “variation”
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Kähler-Ricci flow on horospherical manifold
Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry
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Generalized Kähler-Ricci flow on toric Fano varieties
Topics Perelman entropy, Kähler–Ricci flow, Kähler geometry · Shared terms “entropy functional”
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Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton
Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “gradient Kähler”
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Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons
Topics Kähler–Ricci solitons, Kähler–Ricci flow, Kähler geometry · Shared terms “shrinking Kähler”, “Kähler metrics”
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Singularity Models of Finite-Time Kähler-Ricci Flows
Topics Kähler–Ricci flow, Kähler geometry, Ricci solitons · Same author Frederick Tsz-Ho Fong
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Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones
Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “shrinking Kähler”, “gradient Kähler”
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
- 6A.12Entropy, Information and Diffusion
Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.
- 6B.16Eigenvalue Problems and Perelman’s λ
Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.