Axially symmetric volume constrained anisotropic mean curvature flow

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume in comparison with their initial surface energy, we show that the flow exists for all time. Numerical simulations of the curvature flow are presented.     

A fully discrete numerical scheme for weighted mean curvature flow

Summary. We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates in natural norms. We also present numerical examples which confirm our theoretical results. Received October 2, 2000 / Published online July 25, 2001     

A modified phase field approximation for mean curvature flow with conservation of the volume

This paper is concerned with the motion of a time‐dependent hypersurface ?Ω(t) in ?^d that evolves with a normal velocity where κ is the mean curvature of ?Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation where ε is an approximation parameter, W a double well potential and c_W a constant that depends only on W. We study here a modified version of this equation and we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume‐preserving properties than the traditional nonlocal Allen–Cahn equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Hyperbolic mean curvature flow

In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with dimension larger than 4. We derive nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow. Moreover, we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski space-time.     

Gaussian mean curvature flow

In the Euclidean Space \mathbb Rⁿ⁺¹{\mathbb {R}^{n+1}} with a density e^{e\frac12 n m2 |x|2}, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.     

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

Harnack inequality for the Lagrangian mean curvature flow

If is a Lagrangian manifold immersed into a K?hler-Einstein manifold, nothing is known about its behavior under the mean curvature flow. As a first result we derive a Harnack inequality for the mean curvature potential of compact Lagrangian immersions immersed into . Received March 16, 1997 / Accepted April 24, 1998     

Starshaped hypersurfaces and the mean curvature flow

Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken and Sinestrari [8], valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori bounds are satisfied for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds that are close to them. This gives a classification of the possible singularities for these surfaces in the casen=2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity occurs.     

Singularities of mean curvature flow

Mean curvature flow and its singularities have been paid attention extensively in recent years. The present article reviews briefly their certain aspects in the author's interests.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

Lorentzian Legendrian mean curvature flow

In this paper we generalize the Legendrian mean curvature flow to Lorentzian geometry. More precisely, we study the case, where the ambient manifold is a Lorentzian Sasaki $\eta $ -Einstein manifold. For Legendrian curves we establish convergence results in Theorems 1.1 and 1.2 and we derive estimates for the Legendrian angle for arbitrary dimensions in Theorem 1.3.     

Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

Summary.  We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u _t −Δu+ɛ⁻² f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length'. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u ₀ . In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow. Received April 30, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10 Correspondence to: A. Prohl     

Self-shrinkers for the mean curvature flow in arbitrary codimension

In this paper, we generalize Colding–Minicozzi’s recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere ${bf S}^{n}(\sqrt{2n})$ is the only complete embedded connected $F$ -stable self-shrinker in $\mathbf{R}^{n+k}$ with $\mathbf{H}\ne 0$ , polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in $\mathbf{R}^4$ with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen–Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not $F$ -stable.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

Implicit time discretization for the mean curvature flow equation

In this paper we apply the method of implicit time discretization to the mean curvature flow equation including outer forces. In the framework ofBV-functions we construct discrete solutions iteratively by minimizing a suitable energy-functional in each time step. Employing geometric and variational arguments we show an energy estimate which assures compactness of the discrete solutions. An additional convergence condition excludes a loss of area in the limit. Thus existence of solutions to the continuous problem can be derived. We append a brief discussion of the related Mullins-Sekerka equation.This work was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 256, Bonn     

Starshaped hypersurfaces¶and the mean curvature flow

Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken and Sinestrari [8], valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori bounds are satisfied for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds that are close to them. This gives a classification of the possible singularities for these surfaces in the case n= 2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity occurs. Received: 6 June 1997 / Revised version: 24 October 1997