Mean Curvature Flow Of Graphs Research Articles

A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

Short time existence and smoothness of the nonlocal mean curvature flow of graphs

Short time existence and smoothness of the nonlocal mean curvature flow of graphs

Mean curvature flow of graphs in generalized Robertson–Walker spacetimes with perpendicular Neumann boundary condition

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a generalized Robertson–Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the metric of such a solution is conformal to the one of the leaf of the GRW in asymptotic time. Furthermore, if the initial hypersurface is mean convex, then the evolving hypersurfaces remain mean convex during the flow.

Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang–Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated [Formula: see text]-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins–Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a [Formula: see text]-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch.

Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an L^{infty }_{omega ,x,t} estimate for the gradient and an L^{2}_{omega ,x,t} bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.

Mean curvature flow of graphs with Neumann boundary conditions

In this paper, we study the mean curvature flow of graphs with Neumann boundary conditions. The main aim is to use the maximum principle to get the boundary gradient estimate for solutions. As an application, we obtain the corresponding long time existence for the mean curvature flow of graphs.

Gradient estimates for mean curvature flow with Neumann boundary conditions

We study the mean curvature flow of graphs both with Neumann boundary conditions and transport terms. We derive boundary gradient estimates for the mean curvature flow. As an application, the existence of the mean curvature flow of graphs is presented. A key argument is a boundary monotonicity formula of a Huisken type derived using reflected backward heat kernels. Furthermore, we provide regularity conditions for the transport terms.

Weak solutions for a stochastic mean curvature flow of two-dimensional graphs

We study a stochastically perturbed mean curvature flow for graphs in $${\mathbb {R}}^3$$ over the two-dimensional unit-cube subject to periodic boundary conditions. The stochastic perturbation is a one dimensional white noise acting uniformly in all points of the surface in normal direction. We establish the existence of a weak martingale solution. The proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach. To overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods.

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σϵ collapsing to a subRiemannian metric σ0 as ϵ→0. We establish Ck,α estimates for this flow, that are uniform as ϵ→0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in Capogna et al. (2013) to the total variation flow.

Forced anisotropic mean curvature flow of graphs in relative geometry

The paper is concerned with the graph formulation of forced anisotropic mean curvature flow in the context of the heteroepitaxial growth of quantum dots. The problem is generalized by including anisotropy by means of Finsler metrics. A semi-discrete numerical scheme based on the method of lines is presented. Computational results with various anisotropy settings are shown and discussed.

Sub-Riemannian heat kernels and mean curvature flow of graphs

We introduce a sub-Riemannian analogue of the Bence–Merriman–Osher algorithm (Merriman et al., 1992 [42]) and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L.C. Evans (1993) [27] in the Euclidean setting and is based on new results on the relation between sub-Riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs.

Mean curvature flows and isotopy problems

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collaborators I. Medoys, K. Smoczyk, and M.-P. Tsui.

Mean curvature flow of entire graphs in a half-space with a free boundary

Abstract. We study the mean curvature flow of graphs with prescribed contact angle on a fixed, smooth hyperplane in Euclidean space. We obtain long time existence and convergence to a self similar solution of the mean curvature flow orthogonal to the fixed hyperplane.

Mean curvature flow of graphs in warped products

Let M be a complete Riemannian manifold which either is compact or has a pole, and let φ be a positive smooth function on M . In the warped product M ×φ R, we study the flow by the mean curvature of a locally Lipschitz continuous graph on M and prove that the flow exists for all time and that the evolving hypersurface is C∞ for t > 0 and is a graph for all t. Moreover, under certain conditions, the flow has a well defined limit.

Construction of complete embedded self-similar surfaces under mean curvature flow. Part II

We study the Dirichlet problem associated to the equation for self-similar surfaces under mean curvature flow for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries.

Mean curvature flow with a forcing term in minkowski space

We study the forced mean curvature flow of graphs in Minkowski space and prove longtime existence of solutions. When the forcing term is a constant, we prove convergence to either a constant mean curvature hypersurface or a translating soliton – depending on the boundary conditions at infinity.

Parabolic equations with continuous initial data

The aim of this thesis is to derive new gradient estimates for parabolic equations. The gradient estimates found are independent of the regularity of the initial data. This allows us to prove the existence of solutions to problems that have non-smooth, continuous initial data. We include existence proofs for problems with both Neumann and Dirichlet boundary data. The class of equations studied is modelled on mean curvature flow for graphs. It includes anisotropic mean curvature flow, and other operators that have no uniform non-degeneracy bound. We arrive at similar estimates by three different paths: a 'double coordinate' approach, an approach examining the intersections of a solution and a given barrier, and a classical geometric approach.

Sharp estimates for mean curvature flow of graphs

Sharp estimates for mean curvature flow of graphs are shown and examples are given to illustrate why these are sharp. The estimates improves earlier (non-sharp) estimates of Klaus Ecker and Gerhard Huisken.

Long time behavior of Riemannian mean curvature flow of graphs

In this paper we consider long time behavior of a mean curvature flow of nonparametric surface in Rn, with respect to a conformal Riemannian metric. We impose zero boundary value, and we prove that the solution tends to 0 exponentially fast as t→∞. Its normalization u/supu tends to the first eigenfunction of the associated linearized problem.

Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs

The efficient numerical simulation of the curvature-driven motion of interfaces is an important tool in several free- boundary problems. We treat the case of an interface which is given as a graph. The highly non-linear problem is discretized in space by piecewise linear finite elements. Although the problem is not in divergence form it can be written in a variational form which allows the use of the modern adaptive techniques of finite elements. The time discretization is carried out in a semi-implicit way such that in every time step a linear system with symmetric positive matrix has to be solved. Optimal error estimates are proved for the fully discrete problem under the assumption that the time-step size is bounded by the spatial grid size.