As on arXiv
Abstract
We relate the generalized Ricci curvature of left-invariant generalized metrics on a Lie group with the Ricci curvature of certain metric Lie algebras. As an application, we prove subconvergence of rescaled generalized Ricci flows to expanding generalized Ricci solitons on simply connected Lie groups with positive semi-definite Killing form. Under the same hypotheses, subconvergence of pluriclosed flows to expanding pluriclosed solitons is also shown. We further prove that left-invariant Bismut-Ricci flat metrics arise as critical points for the generalized scalar curvature functional on unimodular Lie groups. In view of this, we establish dynamical stability for the generalized Ricci flow of so-called standard bi-invariant, Bismut-flat metrics on compact, simply connected, semisimple Lie groups and construct examples of dynamically unstable Bismut-flat metrics which are non-standard.
Comments: 36 pages, comments welcome!
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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 homogeneous generalized Ricci flow
Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons · Same authors Elia Fusi, Ramiro A. Lafuente, James Stanfield · Shared terms “Lie groups”, “generalized metrics”, “invariant generalized”
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Generalized scalar curvature and Ricci flow on compact Lie groups
Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Lie groups”, “Bismut”, “left invariant”
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Generalized Ricci flow on aligned homogeneous spaces
Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “flat metrics”, “Bismut”, “Lie groups”
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Bismut Ricci flat manifolds with symmetries
Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut”, “construct examples”
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Positive Hermitian Curvature Flow on special linear groups and perfect solitons
Topics Generalized & G₂ flows, Homogeneous spaces, Stability & uniqueness · Same author James Stanfield · Shared terms “Lie groups”, “Killing form”, “dynamically unstable”
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Bismut Ricci flat generalized metrics on compact homogeneous spaces (including a Corrigendum)
Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “generalized metrics”, “Bismut”, “invariant generalized”
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Dynamical stability of Pluriclosed and Generalized Ricci solitons
Topics Generalized & G₂ flows, Stability & uniqueness, Ricci solitons · Shared terms “pluriclosed”, “dynamical stability”, “dynamically”
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Stability and moduli space of generalized Ricci solitons
Topics Generalized & G₂ flows, Scalar curvature, Stability & uniqueness · Shared terms “Bismut flat”, “critical points”, “Lie”
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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
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 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.