Paper · Apr 29, 2020

Volume growth estimates for Ricci solitons and quasi-Einstein manifolds

Xu Cheng, Ernani Ribeiro, Detang Zhou

math.DGarXiv:2004.14426
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Abstract

In this article, we provide some volume growth estimates for complete noncompact gradient Ricci solitons and quasi-Einstein manifolds similar to the classical results by Bishop, Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. We prove a sharp volume growth estimate for complete noncompact gradient shrinking Ricci soliton. Moreover, we provide upper bound volume growth estimates for complete noncompact quasi-Einstein manifolds with In addition, we prove that geodesic balls of complete noncompact quasi-Einstein manifolds with and have at most exponential volume growth.

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Submitted
Apr 29, 2020
Subjects
math.DG
Ricci solitonsEinstein metricsAlmost & η-Ricci solitonsKähler geometryBoundaries & non-compact

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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 2016, 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 Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature (Xu Cheng, Matheus Vieira and Detang Zhou, 2019)1Cheng 2019 Four-dimensional complete gradient shrinking Ricci solitons (Huai-Dong Cao, Ernani Ribeiro and Detang Zhou, 2020)6Cao 2020 A Study On Some Geometric Properties and Physical Applications of A Mixed Quasi-Einstein Spacetime (Kaushik Chattopadhyay, Arindam Bhattacharyya and Dipankar Debnath, 2021)8Chattopadhya 2021 Rigidity of four-dimensional Gradient shrinking Ricci solitons (Xu Cheng and Detang Zhou, 2021)2Cheng 2021 Volume growth estimates of gradient Ricci solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2022)3Chan 2022 The extended quasi-Einstein manifolds with generalised Ricci solitons (Zhiming Huang, Weijun Lu and Fuhong Su, 2022)4Huang 2022 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)5Bhattacharya 2025 The rigity of spaces with cyclic parallel Ricci tensor (Fengyuan Zhang, 2026)7Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature

    Xu Cheng, Matheus Vieira and Detang Zhou

    Topics Boundaries & non-compact, Ricci solitons · Same authors Xu Cheng, Detang Zhou · Shared terms “volume growth”, “complete noncompact”, “gradient Ricci”

  2. Rigidity of four-dimensional Gradient shrinking Ricci solitons

    Xu Cheng and Detang Zhou

    Topics Boundaries & non-compact, Ricci solitons · Same authors Xu Cheng, Detang Zhou · Shared terms “noncompact gradient”, “complete noncompact”, “gradient shrinking”

  3. Volume growth estimates of gradient Ricci solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Ricci solitons · Shared terms “growth estimate”, “volume growth”, “gradient Ricci”

  4. The extended quasi-Einstein manifolds with generalised Ricci solitons

    Zhiming Huang, Weijun Lu and Fuhong Su

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  5. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “gradient Ricci”

  6. Four-dimensional complete gradient shrinking Ricci solitons

    Huai-Dong Cao, Ernani Ribeiro and Detang Zhou

    Topics Ricci solitons · Same authors Ernani Ribeiro, Detang Zhou · Shared terms “gradient shrinking”, “shrinking Ricci”, “addition”

  7. The rigity of spaces with cyclic parallel Ricci tensor

    Fengyuan Zhang

    Topics Almost & η-Ricci solitons, Einstein metrics, Boundaries & non-compact · Shared terms “quasi Einstein”, “gradient shrinking”, “shrinking Ricci”

  8. A Study On Some Geometric Properties and Physical Applications of A Mixed Quasi-Einstein Spacetime

    Kaushik Chattopadhyay, Arindam Bhattacharyya and Dipankar Debnath

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

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

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

  2. 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.