As on arXiv
Abstract
The generalized Einstein Hilbert action is an extension of the classic scalar curvature energy and Perelman F functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. Through a delicate analysis of the group of generalized gauge transformations, and implementing a novel connection, we give a simple formula for the second variation of this energy which generalizes the Lichnerowicz operator in the Einstein case. As an application, we show that all Bismut flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso and Podesta, Spiro, Kröncke to the moduli space of generalized Ricci solitons. To finish we classify deformations of the Bismut-flat structure on S3 and show that some are integrable while others are not.
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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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The Stability of Generalized Ricci Solitons
Topics Generalized & G₂ flows, Stability & uniqueness, Ricci solitons · Same author Kuan-Hui Lee · Shared terms “Bismut flat”, “generalized Einstein”, “linearly stable”
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Dynamical stability of Pluriclosed and Generalized Ricci solitons
Topics Generalized & G₂ flows, Stability & uniqueness, Ricci solitons · Same author Kuan-Hui Lee · Shared terms “generalized Einstein”, “integrable”, “Hilbert”
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Bochner formulas, functional inequalities and generalized Ricci flow
Topics Perelman entropy, Generalized & G₂ flows · Shared terms “Bismut”
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Homogeneous Generalized Ricci flows II
Topics Generalized & G₂ flows, Scalar curvature, Stability & uniqueness · Shared terms “Bismut flat”, “critical points”, “Lie”
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A Spinorial Perelman's Functional: Critical Points and Gradient Flow
Topics Perelman entropy, Stability & uniqueness, Ricci solitons · Shared terms “critical points”, “variation”
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Nash Entropy and Heat Kernel on Generalized Ricci Flow
Topics Perelman entropy, Generalized & G₂ flows
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Kähler-Ricci flow for deformed complex structures
Topics Perelman entropy, Stability & uniqueness, Ricci solitons · Shared terms “moduli space”, “variation”, “Lie”
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Backward Pseudolocality of -Laplacian Flow
Topics Perelman entropy, Generalized & G₂ flows, Scalar curvature
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.