SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 55 Issue: 5

The Sobolev inequalities along the Ricci-Bourguignon flow and applications

Mehdi Jafari(payame Noor university)0000-0002-7154-7527
Shahroud Azami(‎Imam Khomeini International University)0000-0002-8976-2014
U.c. De(‎University of Calcutta)0000-0002-8990-4609
Pages
—
Publication Date
2026-10-28
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Additionally, {this paper investigates} the $\kappa$-noncollapsing estimates along the RBF.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Consider a closed Riemannian manifold $M$ where the metrics $g(t)$ evolve by the Ricci-Bourguignon flow (RBF).The paper first derives a Sobolev inequality along the RBF.
📊
Key Findings & Data:Then, by applying this Sobolev inequality we obtain a linear parabolic estimate and an upper bound estimate for the heat kernel along the RBF.
💡
Conclusion & Impact:Additionally, {this paper investigates} the $\kappa$-noncollapsing estimates along the RBF.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Ricci-Bourguignon flow · eigenvalue · monotonicity formula

Abstract

Consider a closed Riemannian manifold $M$ where the metrics $g(t)$ evolve by the Ricci-Bourguignon flow (RBF).The paper first derives a Sobolev inequality along the RBF. Then, by applying this Sobolev inequality we obtain a linear parabolic estimate and an upper bound estimate for the heat kernel along the RBF. Additionally, {this paper investigates} the $\kappa$-noncollapsing estimates along the RBF.

🔖

Cite This Article

Jafari, M., Azami, S., De, U. (2026). The Sobolev inequalities along the Ricci-Bourguignon flow and applications. Hacettepe Journal of Mathematics and Statistics, Volume: 55 Issue: 5. https://doi.org/10.15672/hujms.1618538

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (23)

  1. A. Abolarinwa, Sobolev-type inequalities and heat kernel bounds along the geometric flow, Afr. Math. 27, 169–186, 2016.
  2. T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (4), 573–598, 1976.
  3. S. Azami, Monotonicity of eigenvalues of Witten-Laplace operator along the Ricci- Bourguignon flow, AIMS Mathematics 2 (2), 230–243, 2017.
  4. S. Azami, First eigenvalues of geometric operator under the Ricci–Bourguignon flow, J. Indones. Math. Soc. 24 (1), 51–61, 2018.
  5. S. Azami, Eigenvalue monotonicity of (p, q)-Laplacian along the Ricci-Bourguignon flow, Communications of the Korean Mathematical Society 34 (1), 287–301, 2019.
  6. M. Bailesteanu, Bounds on the heat kernel under the Ricci flow, Proc. Am. Math. Soc. 140, 691–700, 2012.
  7. M. Bailesteanu and H. Tran, Heat kernel estimates under the Ricci-harmonic flow, Proceedings of the Edinburgh Mathematical Society 60 (4), 831–857, 2017.
  8. J. P. Bourguignon, Ricci curvature and Einstein metrics, in: Global differential geometry and global analysis (Berlin, 1979), Lecture Notes in Math. 838, 42–63, Springer, Berlin, 1981.
  9. G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza and L. Mazzieri, The Ricci- Bourguignon flow, Pacific J. Math. 287 (2), 337–370, 2017.
  10. B. Chen, Q. He and F. Zeng, Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow, Pacific Journal of Mathematics 296 (1), 1–20, 2018.
  11. S. Fang and T. Zheng, An upper bound of the heat kernel along the harmonic-Ricci flow, Manuscripta Math. 151, 1–18, 2016.
  12. S. Fang and T. Zheng, The (logarithmic) Sobolev inequalities along geometric flow and applications, J. Math. Anal. Appl. 434, 729–764, 2016.
  13. E. Hebey, Optimal Sobolev inequalities on complete Riemannian manifolds with Ricci curvature bounded below and positive injectivity radius, Amer. J. Math. 118 (2), 291–300, 1996.
  14. E. Hebey and M. Vaugon, Meilleures constantes dans le therme d’inclusion de Sobolev, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1), 57–93, 1996.
  15. P. T. Ho, On the Ricci–Bourguignon flow, International Journal of Mathematics 31 (6), 2050044, 2020.
  16. S. Y. Hsu, Uniform Sobolev inequalities for manifolds evolving by Ricci flow, International Mathematics Research Notices 2007 (9), rnm056, 2007.
  17. X. G. Liu and K. Wang, A Gaussian upper bound of the conjugate heat equation along an extended Ricci flow, arXiv: 1412.3200, 2014.
  18. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159, 2002.
  19. H. Qiu and A. Zhu, Ricci-Bourguignon flow on manifolds with boundary, Chin. Ann. Math. Ser. B 42, 953–968, 2021.
  20. L. F. Wang, Monotonicity of eigenvalues and functionals along the Ricci-Bourguignon flow, J. Geom. Anal. 29, 1116–1135, 2019.
  21. R. Ye, Curvature estimates for the Ricci flow I, Calc. Var. 31, 417–437, 2008.
  22. R. Ye, The logarithmic Sobolev and Sobolev inequalities along the Ricci flow, Commun. Math. Stat. 3, 1–36, 2015.
  23. Q. S. Zhang, A uniform Sobolev inequality under Ricci flow, Int. Math. Res. Not. 2007, 1–17, 2007.