Introduction to mean curvature flow

— This is a short overview on the most classical results on mean curvature flow as a flow of smooth hypersurfaces. First of all we define the mean curvature flow as a quasilinear parabolic equation and give some easy examples of evolution. Then we consider the M.C.F. on convex surfaces and sketch the proof of the convergence to a round point. Some interesting results on the M.C.F. for entire graphs are also mentioned. In particular when we consider the case of dimension one, we can compute the equation for the translating graph solution to the curve shortening flow and solve it directly. 1. Notation and definitions We consider an n−dimensional smooth, orientable manifold M and a smooth immersion in Euclidean space F : M → Rn+1. Given local coordinates φ : R →M (x1, . . . , xn) 7→ φ (x1, . . . , xn) we denote by g the metric on F(M) induced by the standard scalar product 〈·, ·〉 of Rn+1 ∀p =φ (x) gij (p) = 〈 ∂F ∂xi (p) , ∂F ∂xj (p) 〉 . The elements of the inverse matrix g = {gij} are also used to arise indices in the Einstein summation convention. Since F (M) is orientable, there exists an outer normal vector field ν on M and we can define the second fundamental form hij (p) = 〈 ∂F ∂xi (p) , ∂ν ∂xj (p) 〉 = − 〈 ∂2F ∂xi∂xj (p) , ν (p) 〉 ,