Surface diffusion flow near spheres

Surface diffusion flow near spheres

The Intermediate Surface Diffusion Flow on Spheres

It is shown that solutions to the intermediate surface diffusion flow are real analytic in space and time, provided the initial surface is real diffeomorphic to a Euclidean sphere.

The Willmore Flow near Spheres

The Willmore flow leads to a quasilinear evolution equation of fourth order. We study existence, uniqueness and regularity of solutions. Moreover, we prove that solutions exist globally and converge exponentially fast to a sphere, provided that they are initially close to a sphere.

On the Surface Diffusion Flow

In this paper we present recent existence, uniqueness, and stability results for the motion of immersed hypersurfaces driven by surface diffusion. We provide numerical simulations for curves and surfaces that exhibit the creation of singularities. Moreover, our numerical simulations show that the flow causes a loss of embeddedness for some initially embedded configurations.

The Volume Preserving Mean Curvature Flow near Spheres

By means of a center manifold analysis we investigate the averaged mean curvature flow near spheres. In particular, we show that there exist global solutions to this flow starting from non-convex initial hypersurfaces.