Related papers: Numerical computation of constant mean curvature s…
The mean curvature flow describes the evolution of a surface (a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical…
We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity…
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An…
Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system \[ \begin{split} &\partial \psi_{1} = (|\psi_{1}|^{2} + |\psi_{2}|^{2})…
We consider a numerical scheme for the approximation of a system that couples the evolution of a two--dimensional hypersurface to a reaction--diffusion equation on the surface. The surfaces are assumed to be graphs and evolve according to…
We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the…
We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing…
We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.
We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the…
We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…
We consider the numerical approximation of axisymmetric mean curvature flow with the help of linear finite elements. In the case of a closed genus-1 surface, we derive optimal error bounds with respect to the $L^2$-- and $H^1$--norms for a…
We consider a finite difference approximation of mean curvature flow for axisymmetric surfaces of genus zero. A careful treatment of the degeneracy at the axis of rotation for the one dimensional partial differential equation for a…
The level-set method is a popular method for interface capturing. One of the advantages of the level-set method is that the curvature and the normal vector of the interface can be readily calculated from the level-set function. However, in…
In the process of projecting the surface of a three-dimensional object onto a two-dimensional surface, due to the perspective distortion, the image on the surface of the object will have different degrees of distortion according to the…
We investigate the gradient flow of the $L^2$ norm of the Riemannian curvature on surfaces. We show long time existence with arbitrary initial data, and exponential convergence of the volume normalized flow to a constant scalar curvature…
We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their…
In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of $|A|^2$, where $|A|^2$ is the norm squared of the…
We present a computational scheme that derives a global polynomial level set parametrisation for smooth closed surfaces from a regular surface-point set and prove its uniqueness. This enables us to approximate a broad class of smooth…
We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in…
A finite element method for solving nonlinear differential equations on a grid, with potential applicability to computational fluid dynamics (CFD), is developed and tested. The current method facilitates the computation of solutions of a…