Gaussian mean curvature flow

In the Euclidean Space mathbb Rⁿ⁺¹{mathbb {R}^{n+1}} with a density e^{efrac12 n m2 |x|2}, (e = ±1){e^{varepsilon frac12 n mu^2 |x|^2},} {(varepsilon =pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.