Forced convex mean curvature flow in Euclidean spaces

We show the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can converge to a round sphere in special cases. Long time existence and convergence of the normalization of the flow are studied.     

Translating Surfaces of the Non-parametric Mean Curvature Flow in Lorentz Manifold M2 × R*

In this paper, for the Lorentz manifold M² × R with M² a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M² , which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.     

Evolution of convex hypersurfaces by powers of the mean curvature

We study the evolution of a closed, convex hypersurface in ℝⁿ⁺¹ in direction of its normal vector, where the speed equals a positive power k of the mean curvature. We show that the flow exists on a maximal, finite time interval, and that, approaching the final time, the surfaces contract to a point. The author was partially supported by a Schweizerische Nationalfonds grant No. 21-66743.01.     

Rate of convergence of the mean curvature flow

We study the flow M_t of a smooth, strictly convex hypersurface by its mean curvature in ?^{n + 1}. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x^* (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere Sⁿ of radius √n. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us that the rate of the exponential decay is at least 2/n. We can define the “arrival time” u of a smooth, strictly convex, n‐dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u(x) = t if x ∈ M_t for x ∈ Int(M₀). Huisken proved that, for n ≥ 2, u(x) is C² near x^*. The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C³‐regularity of u. As a consequence of the obtained rate of convergence of the mean curvature flow, we prove that u is not necessarily C³ near x^* for n ≥ 2. We also show that the obtained rate of convergence 2/n, which arises from linearizing a mean curvature flow, is the optimal one, at least for n ≥ 2. © 2007 Wiley Periodicals, Inc.     

On mean curvature flow with forcing

Abstract

This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow’s exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed by Feldman-Kim to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.     

Surfaces with parallel normalized mean curvature vector

A surfaceM in a Riemannian manifold is said to have parallel normalized mean curvature vector if the mean curvature vector is nonzero and the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. In this paper, it is proved that every analytic surface in a euclideanm-spaceE ^m with parallel normalized mean curvature vector must either lies in aE ⁴ or lies in a hypersphere ofE ^m as a minimal surface. Moreover, it is proved that if a Riemann sphere inE ^m has parallel normalized mean curvature vector, then it lies either in aE ³ or in a hypersphere ofE ^m as a minimal surfaces. Applications to the classification of surfaces with constant Gauss curvature and with parallel normalized mean curvature vector are also given.     

Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝ^{n + 1} remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1 , and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

Null 2-type submanifolds of the Euclidean space E 5 with non-parallel mean curvature vector

Let M be a 3-dimensional submanifold of the Euclidean space E⁵ such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector then the normal bundle is flat. We also prove that M is an open portion of a 3-dimensional helical cylinder if and only if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector.     

A stochastic selection principle in case of fattening for curvature flow

Consider two disjoint circles moving by mean curvature plus a forcing term which makes them touch with zero velocity. It is known that the generalized solution in the viscosity sense ceases to be a curve after the touching (the so-called fattening phenomenon). We show that after adding a small stochastic forcing , in the limit the measure selects two evolving curves, the upper and lower barrier in the sense of De Giorgi. Further we show partial results for nonzero . Received: 3 November 2000 / Accepted: 4 December 2000 / Published online: 23 April 2001     

Volume-preserving flow by powers of the mth mean curvature

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.     

Mean curvature flow with a constant forcing

In this paper, the motion of surface by its mean curvature and a forcing term θ was studied. We show that to each uniformly convex bounded initial surface M ₀, there exists a unique θ ^* such that the surface shrinks or expands depending on whether θ < θ ^* or θ > θ ^*. Also, we show that the surface with θ = θ ^* converges to a limiting surface.     

Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planes

A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of A_H . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP ² as well as in complex hyperbolic plane CH ². We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP ² and 41 families in CH ². All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP ² or in CH ² is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

Bounding dimension of ambient space by density for mean curvature flow

For an ancient solution of the mean curvature flow, we show that each time slice M_t is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient solution is a family M_t that evolves under mean curvature flow for all negative time t.     

Generic Uniqueness for Two Inverse Problems in POtential Scattering

which is written in terms of the characteristic function of the evolving set. The argument is based on implicit time-discretization, derivation of uniform estimates, and passage to thIn this paper the equation of mean curvature flow (with forcing term) is modified, to account not only for surface motion but also for nucleation and other discontinuities in set evolution. Existence of a solution is proved for a weak formulation,e limit.     

Longtime existence of the Lagrangian mean curvature flow

Given a compact Lagrangian submanifold in flat space evolving by its mean curvature, we prove uniform -bounds in space and C²-estimates in time for the underlying Monge-Ampére equation under weak and natural assumptions on the initial Lagrangian submanifold. This implies longtime existence and convergence of the Lagrangian mean curvature flow. In the 2-dimensional case we can relax our assumptions and obtain two independent proofs for the same result.Received: 3 September 2002, Accepted: 12 June 2003, Published online: 4 September 2003Mathematics Subject Classification (2000): 53C44     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

The $epsiv;-Regularized Mean Curvature Flow in One Dimension Space

The evolutionary motion of surfaces or curves by their meancurvature has found much interest during the last years. The problem withmean curvature flow is that singularities can appear during the evolutioneven if the initial surface is convex. To prove the existence of a viscositysolution u of the mean curvature flow, Evans and Spruck [4] builtthe -regularized mean curvature flow. For practicalpurposes, i.e., numerical computations, it would be interesting to knowhow fast the solution of the regularized problem converges to the viscocitysolution of the original problem. The goal in this paper is to presentsome results concerning the -regularized mean curvatureflow in the one-dimension space. It is proved that there exists anasymptotic expansion of the solution of the regularized problem, in powers of theparameter , such that the first term of the asymptoticexpansion is the viscosity solution of the mean curvature flow problem.Moreover, that this asymptotic expansion is true in appropriate topologies,in particular in weighted Sobolev spaces is proved. Finally, an estimateof the rate of convergence in these topologies is given.