Stability of spheres under volume-preserving mean curvature flow (Q536113)

A new elementary proof of the theorem due to \textit{J. Escher} and \textit{G. Simonett} [Proc. Am. Math. Soc. 126, No. 9, 2789--2796 (1998; Zbl 0909.53043)] is given. It says that for the volume-preserving mean curvature flow, if the initial data is close to a Euclidean sphere, then it converges to a Euclidean sphere. An orthogonal decomposition of the solution near the manifold of Euclidean spheres and differential inequalities for the Lyapunov functionals are applied.