The Surface Diffusion Flow for Immersed Hypersurfaces

Abstract

Abstract. We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diffusion. If the initial surface is embedded and close to a sphere, we prove that the solution exists globally and converges exponentially fast to a sphere. Furthermore, we provide numerical simulations showing the creation of singularities for immersed curves.

A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow

“…The time step for the experiment was 10 −3 , with spatial distances of the vertices ranging from 0.01 to 0.05. Also of interest is the decrease of the total quadratic curvature Γ(t) H 2 dµ, see (6). The integral levels off at about 19.73878, which is merely about −0.02% away from the theoretical minimum of 2π 2 , see Figure 3.…”

“…It is interesting to compare the Willmore flow with the surface diffusion flow given by f = ∆H. In the latter case it is well-known that hypersurfaces evolve in such a way as to reduce surface area, while preserving the volume enclosed by Γ(t), see [6] for instance. It is therefore of interest to compare and contrast the behavior of solutions for the Willmore flow and the surface diffusion flow, as will be done in some of the subsequent numerical simulations.…”

Self-Intersections for Willmore Flow

“…Our approach relies on results and techniques in [6,12,16], and we follow closely the original argument in [12]. Lastly we mention that numerical simulations [13] seem to indicate that the Willmore flow can drive immersed surfaces to topological changes in finite time.…”

Diffeomorphic Surface Flows: A Novel Method of Surface Evolution

“…In the last decade, there have been studies on geometric flows such as the celebrated mean curvature flow, e.g., [12] and references therein, surface diffusion flow, e.g., [13], and the Willmore flow, e.g., [27,26]. The equations for these surface flows are second or fourth order parabolic PDEs which require sophisticated numerical methods, e.g., [9].…”