Volume preserving mean curvature flow of L2-almost umbilical hypersurfaces in hyperbolic space

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an [Formula: see text]-almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)].