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

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.

کلیدواژه ها:

Sobolev constant ، Gradient estimate ، Ricci soliton ، Bakry-{\'E}mery Ricci curvature