International Journal of Nonlinear Analysis and Applications

Gradient estimate of equations with potential under the almost Ricci soliton condition

Articles in Press

Document Type : Research Paper

Authors

1 Department of pure mathematics, Faculty of science, Imam Khomeini International University, Qazvin, Iran

2 Department of Pure Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran

Abstract

Using volume comparison theorems and the Sobolev inequality with almost Ricci solitons, we study an important version of the gradient estimate for the solutions of $\Delta u=f+Hu$, for some function $f$, $H$, and we obtain an upper bound for the gradient of $u$ on almost Ricci solitons.

Keywords

[1] S. Azami and S. Hajiaghasi, New volume comparison with almost Ricci soliton, Commun. Korean Math. Soc. 37 (2022), no. 3, 839–849.
[2] R.H. Bamler, Entropy and heat kernel bounds on a Ricci flow background, arXiv:2008.07093v3 [math.DG] (2021).
[3] A. Barros and E. Ribeiro Jr, Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc. 140 (2012), no. 3, 1033–1040.
[4] X. Cao and R.S. Hamilton, Differential Harnack estimates for time-dependent heat equations, Geom. Funct. Anal. 19 (2009), no. 4, 989–1000.
[5] X. Dai, G. Wei, and Z. Zhang, Local Sobolev constant estimate for integral Ricci curvature bounds, Adv. Math. 325 (2018), 1–33.
[6] S. Deshmukh, Almost Ricci solitons isometric to spheres, Int. J. Geometric Meth. Mod. Phys. 16 (2019), no. 5 1950073.
[7] S. Deshmukh and H. Al-Sodais, A note on almost Ricci soliton, Anal. Math. Phys. 10 (2020), 1–11.
[8] S. Deshmukh, H. Alsodais, and N. Bin Turki, Some results on Ricci almost solitons, Symmetry 13 (2021), 430.
 [9] Q. Han and F. Lin, Elliptic Partial Differential Equations, American Mathematical Society, 1997.
[10] W. Hebisch and L. Saloff-Costa, On the relation between elliptic and parabolic Harnack inequalities Ann. Inst. Fourier 51 (2001), no. 5, 1437–1481.
[11] A. Ghosh, Ricci almost solitons satisfying certain conditions on the potential vector field, Publ. Math. Debrecen 87 (2015), no. 1-2, 103–110.
[12] P. Petersen and G.F. Wei, Analysis and geometry on manifolds with integral Ricci curvature bounds II, Trans. Amer. Math. Soc. 353 (2000), no. 2, 457–478.
[13] S. Pigola, M. Rigoli, M. Rimoldi, and A.G. Setti, Ricci almost solitons, Ann. Scuola Normale Super. Pisa-Classe Sci. 10 (2011), no. 4, 757–799.
[14] C. Rose, Heat kernel upper bound on Riemannian manifolds with locally uniform Ricci curvature integral bounds, J. Geom. Anal. 27 (2017), no. 2, 1737–1750.
[15] R. Sharma, Almost Ricci solitons and K-contact geometry, Monatsh. Math. 175 (2014), 621–628.
[16] L. Wang and G. Wei, Local Sobolev constant estimate for integral Bakry-Emery Ricci curvature, Pacific J. Math. 300 (2019), no. 1, 233–256.
[17] G. Wei, R. Ye, A Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds, Journal of Geometric Analysis 19 (2009), no. 3, 719-736.
[18] Q.S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Not. 2006 (2006), Art.ID92314.
[19] Q.S. Zhang and M. Zhu, New volume comparison results and applications to degeneration of Riemannian metrics, Adv. Math. 352 (2019), 1096–1154.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 02 June 2025
  • Receive Date: 11 November 2022
  • Revise Date: 12 October 2023
  • Accept Date: 22 December 2023