Paper · Jan 12, 2022

Spinors and mass on weighted manifolds

Julius Baldauf, Tristan Ozuch

math.DGarXiv:2201.04475v2
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Abstract

This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow.

Comments: 19 pages; final version published in Comm. Math. Phys

Ricci flowScalar curvatureEigenvalues & spectrumPhysics

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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.

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 The spinorial energy for asymptotically Euclidean Ricci flow (Julius Baldauf and Tristan Ozuch, 2022)1Baldauf 2022 Scalar curvature, entropy, and generalized Ricci flow (Jeffrey Streets, 2022)8Streets 2022 ADM mass for C⁰ metrics and distortion under Ricci-DeTurck flow (Paula Burkhardt-Guim, 2022)4Burkhardt-Gu 2022 Harmonic spinors in the Ricci flow (Julius Baldauf, 2022)5Baldauf 2022 Ancient and expanding spin ALE Ricci flows (Isaac M. Lopez and Tristan Ozuch, 2024)6Lopez 2024 Linear stability and instability of Kähler Ricci solitons (Keaton Naff and Tristan Ozuch, 2025)7Naff 2025 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)3Lee 2026 A continuous Positive Mass Theorem for perturbations of Euclidean space (Paula Burkhardt-Guim, 2026)2Burkhardt-Gu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The spinorial energy for asymptotically Euclidean Ricci flow

    Julius Baldauf and Tristan Ozuch PDF

    Topics Ricci flow · Same authors Julius Baldauf, Tristan Ozuch · Shared terms “asymptotically Euclidean”, “spinorial”, “Dirichlet energy”

  2. A continuous Positive Mass Theorem for perturbations of Euclidean space

    Paula Burkhardt-Guim PDF

    Topics Physics, Scalar curvature · Shared terms “mass theorem”, “ADM mass”, “Euclidean”

  3. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon PDF

    Topics Physics, Scalar curvature · Shared terms “ADM mass”, “mass theorem”, “Euclidean”

  4. ADM mass for metrics and distortion under Ricci-DeTurck flow

    Paula Burkhardt-Guim PDF

    Topics Physics, Scalar curvature · Shared terms “ADM mass”

  5. Harmonic spinors in the Ricci flow

    Julius Baldauf PDF

    Topics Ricci flow · Same author Julius Baldauf · Shared terms “Witten”, “gradient flow”, “Perelman”

  6. Ancient and expanding spin ALE Ricci flows

    Isaac M. Lopez and Tristan Ozuch PDF

    Topics Scalar curvature · Same author Tristan Ozuch · Shared terms “weighted mass”, “Perelman”

  7. Linear stability and instability of Kähler Ricci solitons

    Keaton Naff and Tristan Ozuch PDF

    Topics Eigenvalues & spectrum · Same author Tristan Ozuch

  8. Scalar curvature, entropy, and generalized Ricci flow

    Jeffrey Streets PDF

    Topics Physics, Scalar curvature · Shared terms “weighted scalar”, “associated weighted”, “monotonicity formula”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.