Paper · Aug 26, 2026

Topology of low-dimensional generalized Ricci solitons and string backgrounds

Jeffrey Streets

math.DGarXiv:2608.25829
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Abstract

Adapting ideas of, we show that compact generalized Ricci solitons (GRS) have positive Yamabe invariant. We observe a Cheeger-Gromoll-type splitting theorem for GRS as a corollary of the splitting theorem for Bakry-Émery Ricci curvature in. Using this we show that low dimensional GRS are diffeomorphic to or . We determine various topological constraints on string backgrounds (Bismut-Hermitian-Einstein (BHE), strong torsion , strong torsion -manifolds) and show in most cases that they cannot exist on the same manifolds as their classical special holonomy counterparts. Finally we determine the topology of BHE threefolds under natural constraints, relying on an extension of parts of Kollar's characterization of Seifert fibered -manifolds over complex orbifolds.

Generalized & G₂ flowsRicci solitonsRicci-flat & Calabi–Yau3-manifoldsLimit spaces & RCD

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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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)3Gindi 2021 Bochner formulas, functional inequalities and generalized Ricci flow (Eva Kopfer and Jeffrey Streets, 2022)5Kopfer 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)4Apostolov 2022 Optimal Transport and Generalized Ricci Flow (Eva Kopfer and Jeffrey Streets, 2023)2Kopfer 2023 The Riemannian curvature identities of a G₂ connection with skew-symmetric torsion and generalized Ricci solitons (Stefan Ivanov and Nikola Stanchev, 2023)6Ivanov 2023 The canonical symmetry reduction of string backgrounds (Aaron Kennon and Jeffrey Streets, 2025)1Kennon 2025 Pluriclosed flow on Oeljeklaus-Toma manifolds (Jeffrey Streets and Xiaokang Wang, 2025)7Streets 2025 Sasaki with torsion manifolds and string backgrounds (Beatrice Brienza, Anna Fino, Udhav Fowdar et al., 2026)8Brienza 2026
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  1. The canonical symmetry reduction of string backgrounds

    Aaron Kennon and Jeffrey Streets PDF

    Topics Generalized & G₂ flows, Ricci solitons · Same author Jeffrey Streets · Shared terms “string backgrounds”, “holonomy”, “Hermitian”

  2. Optimal Transport and Generalized Ricci Flow

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    Topics Generalized & G₂ flows, Limit spaces & RCD · Same author Jeffrey Streets · Shared terms “determine”

  3. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “dimensional generalized”

  4. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al. PDF

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “Bismut”

  5. Bochner formulas, functional inequalities and generalized Ricci flow

    Eva Kopfer and Jeffrey Streets PDF

    Topics Generalized & G₂ flows · Same author Jeffrey Streets · Shared terms “determine”, “Bismut”, “characterization”

  6. The Riemannian curvature identities of a connection with skew-symmetric torsion and generalized Ricci solitons

    Stefan Ivanov and Nikola Stanchev PDF

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Ricci solitons

  7. Pluriclosed flow on Oeljeklaus-Toma manifolds

    Jeffrey Streets and Xiaokang Wang PDF

    Topics Generalized & G₂ flows, Limit spaces & RCD · Same author Jeffrey Streets

  8. Sasaki with torsion manifolds and string backgrounds

    Beatrice Brienza, Anna Fino, Udhav Fowdar et al. PDF

    Topics Generalized & G₂ flows · Shared terms “string backgrounds”, “Bismut”, “Hermitian”

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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.

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    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.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.