Paper · Jan 5, 2026

A Spinorial Perelman's Functional: Critical Points and Gradient Flow

Tsz-Kiu Aaron Chow, Frederick Tsz-Ho Fong

math.DGarXiv:2601.01863
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Abstract

In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L^2-gradient flow of this functional, and establish its short-time existence and uniqueness via contraction mapping methods.

Comments: 23 pages. Comments are welcome!

Ricci solitonsPerelman entropyStability & uniquenessHarmonic maps & DeTurck trick

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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.

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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

    Shared terms “spinorial”, “critical points”, “spin manifolds”

  2. Stability and moduli space of generalized Ricci solitons

    Kuan-Hui Lee PDF

    Topics Perelman entropy, Stability & uniqueness, Ricci solitons · Shared terms “critical points”, “variation”

  3. Perelman's functionals on manifolds with non-isolated conical singularities

    Xianzhe Dai and Changliang Wang PDF

    Topics Perelman entropy · Shared terms “Perelman functionals”

  4. Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow

    Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al. PDF

    Topics Perelman entropy, Harmonic maps & DeTurck trick

  5. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  6. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash PDF

    Topics Stability & uniqueness, Ricci solitons · Shared terms “critical points”

  7. Stochastic Thermodynamics on Time-Evolving Curved Spaces

    Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al. PDF

    Topics Perelman entropy

  8. A local Sobolev inequality on Ricci flow and its applications

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Perelman entropy

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 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) Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    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.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    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.