This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

Computation of geometric partial differential equations and mean curvature flow

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R 3 is at least 2 2 . We prove this conjecture using the min-max theory of minimal surfaces.

Min-Max theory and the Willmore conjecture

We investigate point singularities of Willmore surfaces, which for example appear as blowups of the Willmore flow near singularities, and prove that closed Willmore surfaces with one unit density point singularity are smooth in codimension one. As applications we get in codimension one that the Willmore flow of spheres with energy less than 8π exists for all time and converges to a round sphere and further that the set of Willmore tori with energy less than

http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n1-p07.pdf

Removability of point singularities of Willmore surfaces

We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovasz–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-Emery.

On the role of convexity in isoperimetry, spectral gap and concentration

A new formulation for the Euler–Lagrange equation of the Willmore functional for immersed surfaces in ℝ
 m
 is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting $\vec{H}$
 be the mean curvature vector of the surface, our new formulation reads ${\mathcal{L}}\vec{H}=0$
 , where $\mathcal{L}$
 is a well-defined locally invertible self-adjoint elliptic operator. Several consequences are studied. In particular, the long standing open problem asking for a meaning to the Willmore Euler–Lagrange equation for immersions having only L
 2-bounded second fundamental form is now solved. The regularity of weak Willmore immersions with L
 2-bounded second fundamental form is also established. Its proof relies on the discovery of conservation laws which are preserved under weak convergence. A weak compactness result for Willmore surfaces with energy less than 8π (the Li–Yau condition ensuring the surface is embedded) is proved, via a point removability result established for Wilmore surfaces in ℝ
 m
 , thereby extending to arbitrary codimension the main result in [KS3]. Finally, from this point-removability result, the strong compactness of Willmore tori below the energy level 8π is proved both in dimension 3 (this had already been settled in [KS3]) and in dimension 4.

/pdf/analysis-aspects-of-willmore-surfaces-5b360gfqam.pdf

Analysis aspects of Willmore surfaces

We consider the L2 gradient flow for the Willmore functional. In [5] it was proved that the curvature concentrates if a singularity develops. Here we show that a suitable blowup converges to a nonumbilic (compact or noncompact) Willmore surface. Furthermore, an L∞ estimate is derived for the tracefree part of the curvature of a Willmore surface, assuming that its L2 norm (the Willmore energy) is locally small. One consequence is that a properly immersed Willmore surface with restricted growth of the curvature at infinity and small total energy must be a plane or a sphere. Combining the results we obtain long time existence and convergence to a round sphere if the total energy is initially small.

/pdf/the-willmore-flow-with-small-initial-energy-4qlob3122g.pdf

The Willmore Flow with Small Initial Energy

We consider curves in ${\mathbb R}^n$ moving by the gradient flow for elastic energy, i.e., the L2 integral of curvature. Long-time existence is proved in the two cases when a multiple of length is added to the energy or the length is fixed as a constraint. Along these lines, a lower bound for the lifespan of solutions to the curve diffusion flow is observed. We derive algorithms for both the elastic flows and the curve diffusion equation. After a numerical test we compute several examples, including cases of curve diffusion in which a singularity develops.

Evolution of Elastic Curves in $\Rn$: Existence and Computation

We consider two-dimensional, compact immersed surfaces in R moving by the gradient of the L integral of their curvature. It is not known whether solutions to this fourth-order, geometric evolution equation may develop singularities in finite time. We give a lower bound on the lifespan of a smooth solution, which depends only on how much the curvature of the initial surface is concentrated in space.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2002/0010/0002/CAG-2002-0010-0002-a004.pdf

Ernst Kuwert

Papers

Removability of point singularities of Willmore surfaces

The Willmore Flow with Small Initial Energy

Evolution of Elastic Curves in $\Rn$: Existence and Computation

Gradient flow for the Willmore functional

Existence of minimizing Willmore surfaces of prescribed genus