Paper · Oct 29, 2025

The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe

Severin Bunk, Miguel Pino Carmona, C. S. Shahbazi

math.DGarXiv:2510.25888
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Abstract

We prove well-posedness of the analytic Cauchy problem for gradient generalized Ricci solitons on an abelian bundle gerbe and solve the initial data equations on every compact Riemann surface. Along the way, we provide a novel characterization of the self-similar solutions of the generalized Ricci flow by means of families of automorphisms of the underlying abelian bundle gerbe covering families of diffeomorphisms isotopic to the identity.

Comments: 29 pages. Comments welcome!

Ricci flowGeneralized & G₂ flowsRicci solitonsSurfaces

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    Miguel Pino Carmona PDF

    Topics Generalized & G₂ flows, Ricci solitons · Same author Miguel Pino Carmona · Shared terms “gradient generalized”

  2. The Stability of Generalized Ricci Solitons

    Kuan-Hui Lee PDF

    Topics Generalized & G₂ flows, Ricci solitons · Shared terms “gradient generalized”

  3. Determinants of pseudo-laplacians and for spinor bundles over Riemann surfaces

    Alexey Kokotov and Dmitrii Korikov PDF

    Topics Surfaces · Shared terms “Riemann surface”, “compact Riemann”

  4. The Heterotic-Ricci flow and its three-dimensional solitons

    Andrei Moroianu, Ángel J. Murcia and C. S. Shahbazi PDF

    Topics Generalized & G₂ flows · Same author C. S. Shahbazi

  5. On the shrinking solitons of generalized Ricci flow

    Xilun Li and Yanan Ye PDF

    Topics Generalized & G₂ flows, Ricci solitons

  6. Dynamical stability of Pluriclosed and Generalized Ricci solitons

    Kuan-Hui Lee PDF

    Topics Generalized & G₂ flows, Ricci solitons

  7. Smoothing a measure on a Riemann surface using Ricci flow

    Peter M. Topping and Hao Yin PDF

    Topics Surfaces, Ricci solitons · Shared terms “Riemann surface”, “initial data”

  8. Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

    Huibin Chen, Zhiqi Chen and Fuhai Zhu PDF

    Topics Generalized & G₂ flows, Ricci solitons

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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 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.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.