Paper · Mar 22, 2026

Asymptotic Geometry of Four-Dimensional Steady Solitons

Aprameya Girish Hebbar, Natasa Sesum

math.DGarXiv:2603.21397v2
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Abstract

In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak -solution and that the soliton is not isometric to the Bryant soliton. In this setting, we identify the two edges of the soliton and prove that the scalar curvature decays at a linear rate away from these edges. Moreover, if the scalar curvature vanishes at infinity, then a stronger inequality holds and the asymptotic cone is a ray. In particular, our results apply to the four-dimensional steady solitons constructed by Lai.

Ricci flowRicci solitonsAncient solutions4-manifoldsScalar curvatureBoundaries & non-compact

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  1. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu PDF

    Topics 4-manifolds, Scalar curvature, Boundaries & non-compact · Shared terms “four dimensional”, “asymptotic geometry”, “complete noncompact”

  2. Unique Asymptotics of Steady Ricci Solitons with Symmetry

    Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum PDF

    Topics 4-manifolds, Ancient solutions, Ricci solitons · Same author Natasa Sesum · Shared terms “curvature decay”

  3. -solutions with the round cylinder as an asymptotic shrinker

    Aprameya Girish Hebbar PDF

    Topics Ancient solutions, Boundaries & non-compact, Ricci solitons · Same author Aprameya Girish Hebbar · Shared terms “steady solitons”, “Bryant”

  4. -symmetry of four-dimensional steady Ricci solitons

    Hanbing Fang PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “dimensional steady”, “four dimensional”

  5. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda PDF

    Topics Ancient solutions, Scalar curvature, Boundaries & non-compact · Shared terms “steady solitons”, “Bryant soliton”, “steady gradient”

  6. Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

    Huai-Dong Cao and Tianbo Liu PDF

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “four dimensional”, “asymptotic cone”

  7. On a dichotomy of the curvature decay of steady Ricci soliton

    Pak-Yeung Chan and Bo Zhu PDF

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “curvature decay”, “four dimensional”, “complete noncompact”

  8. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al. PDF

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Same author Natasa Sesum · Shared terms “four dimensional”, “complete noncompact”

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) 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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

  1. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.