We consider a free boundary problem for the Willmore functional W(f ) = 1 4 Σ H 2 dµ f . Given a smooth bounded domain Ω ⊂ R 3 , we construct Willmore disks which are critical in the class of surfaces meeting ∂Ω at a right angle along their boundary and having small prescribed area. Using rescaling and the implicit function theorem, we first obtain constrained solutions with prescribed barycenter on ∂Ω. We then study the variation of that barycenter.

We study the evolution of a closed hypersurface of the euclidean space by a flow whose speed is given by a power of the scalar curvature. We prove that, if the initial shape is convex and satisfies a suitable pinching condition, the solution shrinks to a point in finite time and converges to a sphere after rescaling. We also give an example of a nonconvex hypersurface which develops a neckpinch singularity.

We study the evolution of a closed immersed hypersurface whose speed is given by a function φ(H ) of the mean curvature asymptotic to H/ ln H for large H . Compared with other nonlinear functions of the curvatures, this speed has some good properties which allow for an easier study of the formation of singularities in the nonconvex case. We prove apriori estimates showing that any surface with positive mean curvature at the initial time becomes asymptotically convex near a singularity. Similar estimates also hold for the mean curvature flow; for the flow considered here they admit a simpler proof based only on the maximum principle.

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Abstract:We consider the evolution of an entire convex graph in euclidean space with speed given by a symmetric function of the principal curvatures. Under suitable assumptions on the speed and on the initial data, we prove that the solution exists for all times and it remains a graph. In addition, after appropriate rescaling, it converges to a homothetically expanding solution of the flow. In this way, we extend to a class of nonlinear speeds the well known results of Ecker and Huisken for the mean curvature flow. MSC: 53C44

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.