Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick
59
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows.
rate research
Read More
We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set $Psubset mathbb R^2$ of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and $P$ is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between ACSF and HCS generalizes the link between ACSF and convex-layer decomposition (Eppstein et al., 2017; Calder and Smart, 2020), which is restricted to convex curves. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.
In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call grid peeling) is the convex-layer decomposition of subsets $Gsubset mathbb Z^2$ of the integer grid, previously studied for the particular case $G={1,ldots,m}^2$ by Har-Peled and Lidicky (2013). The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. (1993) and Sapiro and Tannenbaum (1993). We present empirical evidence that, in a certain well-defined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where $G=mathbb N^2$ is a quarter-infinite grid. This case corresponds to ACSF starting with an infinite L-shaped curve, which when transformed using the ACSF becomes a hyperbola for all times $t>0$. We prove that, in the grid peeling of $mathbb N^2$, (1) the number of grid points removed up to iteration $n$ is $Theta(n^{3/2}log n)$; and (2) the boundary at iteration $n$ is sandwiched between two hyperbolas that are separated from each other by a constant factor.
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 of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.
In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem which reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results of Liu and Terng for the mean curvature flow of isoparametric submanifolds in spheres.
In this paper we investigate the singularities of Lagrangian mean curvature flows in $mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asymptotic to special Lagrangian cones; hence all type II singularities are modeled by unions of special Lagrangian cones.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
