Yoshikazu Giga, Norbert Požár
We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.
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We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.
An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The comparison principle is obtained by means of a suitable weak formulation of the flow, while the existence of a global-in-time solution follows via a minimizing movements approach.
In this work we consider the global existence of volume-preserving crystalline curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address global existence and regularity of the flow for smooth anisotropies. For the non-smooth case we establish global existence results for the types of anisotropies known to be globally well-posed.
A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the va…
In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows to find explicitly the translators and describe their main features.
Discrete-to-continuum crystalline curvature flows
针对晶体平均曲率流，本文提出一种基于新离散距离的完全离散隐式变分格式，以消除空间漂移与钉扎，并证明其在任意维度和广泛纯晶体各向异性下的严格收敛性。
We show that every bounded subset of an Euclidean space can be approximated by a set that admits a certain vector field, the so-called Cahn-Hoffman vector field, that is subordinate to a given anisotropic metric and has a square-integrable divergence. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner. We use this approximation to construct test functions necessary to prove the comparison principle for viscosity solutions of the level set formulation of the crystalline mean curvature flow that were recently in…
We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit.
Area-preserving crystalline curvature flow in two dimensions
研究保面积晶体曲率流下平面集的演化，证明规则初始数据的平流解与经典ODE演化一致，且任意初始数据的平流指数收敛到Wulff形状的不交并，并在非气泡假设下最终正则化。
Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions
该研究针对单一六角各向异性平面网络的晶体曲率流，证明了宽初始条件下经典解的局部存在性，分类了单有界分支的位似收缩解，并给出网络收缩为二重线段的实例。
We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard--like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.
On the self-similar solutions of the crystalline mean curvature flow in three dimensions
本文提出两种正亏格的三维晶面均曲率流自相似收缩解，用于测试基于Chambolle最小化运动方案的三维晶面均曲率流水平集算法数值实现，并展示了有限元方法在处理三维边界的改进效果。
Weak solutions of Anisotropic (and crystalline) inverse mean curvature flow as limits of $p$-capacitary potentials
本文在极温和的各向异性与初始数据条件下，通过 p-容量函数逼近构造了各向异性逆平均曲率流的弱解，证明了解的唯一性、周长增长及正则性等经典性质，并给出了显式解。
We consider the evolution by crystalline curvature of a planar set in a stratified medium, modeled by a periodic forcing term. We characterize the limit evolution law as the period of the oscillations tends to zero. Even if the model is very simple, the limit evolution problem is quite rich, and we discuss some properties such as uniqueness, comparison principle and pinning/depinning phenomena.
In this paper, we derive a new model for the description of liquid crystalline flows. While microscopic Doi type models suffer from the high dimensionality of the underlying product space, the more macroscopic Ericksen--Leslie type models describe only the long time behavior of the flow and are valid only close to equilibrium. By applying an energetic variational approach, we derive a new macroscopic model which shall provide an improved description far from equilibrium. The novelty of our approach lies in the way the energy is minimized. Distinguishing between the velocities of particles and …
Existence and uniqueness for anisotropic and crystalline mean curvature flows
本文研究了带有强迫项和任意凸动力学的晶体曲率流的存在性和唯一性，通过引入新的解定义，证明了在任意维度和初始闭集的情况下，解满足比较原理和稳定性，且处理了时间依赖的有界强迫项。
Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions
本文研究多边形曲线的晶体弹性流，证明其唯一性、可有限次重启以全局演化，并利用Lojasiewicz-Simon不等式证明收敛到稳态解，且给出方形各向异性下稳态解的分类。
A distributional approach to nonlocal curvature flows
将分布方法扩展到非局部曲率流，包括分数阶平均曲率流和Minkowski流，提供适定性公式。
We prove short-time existence of ϕ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.
We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.
A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main applications of the compactness theorem are discussed. First, we obtain a description of critical points/local minimizers of elliptic energies interacting with a confinement potential. Second, we prove an Alexandrov-type theorem for crystalline isoperimetric problems.
We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching condition, then this is preserved and the flow converges to a sphere under rescaling.
We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval $[0,T)$ can be extended over time $T$. Moreover, we show that the condition is optimal in some sense.
We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.
We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow with one additional dimension.
Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature and the mean curvature vanish. Using integral method, we prove global existence and convergence for the $Q$-curvature flow (resp $T$-curvature flow) to smooth metric of prescribed $Q$-curvature (resp $T$-curvature) under conformally invariant assumptions.
We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean…
It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric st…
In [5], Sáez and Schnürer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature flow. In this paper, we generalize their results to a general curvature setting. Our key ingredient is the existence result of general curvature flow with boundary conditions, which is proved in Section 4.
Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists for all time and converges pointwisely to a hyperbolic metric
In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in $R^{n+1}$. We study the basic properties, such as positivity preserving property, of the translating mean curvature flow. The Dirichlet problem for the graphical translating mean curvature flow is studied and the global existence of the flow and the convergence property are also considered.
Evolution of crystalline thin films by evaporation and condensation in three dimensions
该研究在SDRI理论框架下，建立耦合准静态弹性问题的四阶各向异性平均曲率型薄膜演化方程，通过极小化运动方法证明周期边界下有限时间正则解的存在性。
We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow
本文研究带孤立锥形奇点的空间上的Ricci de Turck流，证明其保持Ricci曲率初始正则性与附加假设下的数量曲率正性，为奇异Ricci流的曲率算子正性研究奠定基础。
We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.
Mean curvature flow solitons from symmetry group viewpoint
确定了一般环境Riemannian流形中平均曲率流的对称群，基于此定义了广义孤立子，并给出非欧几里得曲面上的位似孤立子实例，证明所有仿射解均为自相似解。
In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension $d\geq1$, which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to Le-Šešum \cite{LS} and the authors \cite{XYZ1,XYZ2}. Using the extension theorem, we prove two convergence theorems for the mean curvature flow of closed submanifolds in ${R}^{n+d}$ under suitable integral curvature conditions.
We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.
Ancient mean curvature flows with finite total curvature
本文在具有有限全曲率和Morse指标$I$的极小超曲面上构造了$I$-族古代图形平均曲率流，并证明其具有有限全曲率和有限质量下降，其中一族为平均凸。
In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $\mathbb{R}^N$ in every dimension $N \ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension $N \ge 8$, a nontrivial subset of these solutions has uniformly bounded mean curvature.
We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C^{1,1}-regular. We provide the same result also for the volume preserving fractional mean curvature flow.
We prove the existence of closed convex ancient solutions to curvature flows which become more and more oval for large negative times. The speed function is a general symmetric function of the principal curvatures, homogeneous of degree greater than one. This generalises previous work on the mean curvature flow and other one-homogeneous curvature flows. As an auxiliary result, we prove a new theorem on the convergence to a round point of convex rotationally symmetric hypersurfaces satisfying a suitable constraint on the curvatures.
In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in \cite{St} coincide for any Type I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex Type I Mean curvature flow.…
It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. We show that the mean curvature blows up at the singularities of the mean curvature flow starting from an immersed closed hypersurface with small L^2-norm of the traceless second fundamental form (observe that the initial hypersurface is not necessarily convex). As a consequence of the proof of this result we also obtain the dynamic stability of a sphere along the mean curvature flow with respect to the L^2-norm.
We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volume preserving curvature flows. Furthermore we show that these flows converge exponentially in the C-infinity topology to a geodesic sphere.
In this note, we first prove that the solution of mean curvature flow on a finite time interval $[0,T)$ can be extended over time $T$ if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval $[0,T)$ can be extended over time $T$ if the space-time integration of the mean curvature is finite. Moreover, we show that these conditions are optimal in some sense.
In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$ with positive mean curvature is $κ$-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.
In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges to a metric with the prescribed function $f$ as its scalar curvature.
In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.
We show that mean curvature flow of a compact submanifold in a complete Riemannian manifold cannot form singularity at time infinity if the ambient Riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions .
In this paper we analyze the behavior of the distance function under Ricci flows whose scalar curvature is uniformly bounded. We will show that on small time-intervals the distance function is $\frac12$-Hölder continuous in a uniform sense. This implies that the distance function can be extended continuously up to the singular time.
We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a strong uniqueness theorem when the cone is generic and use it to classify mean curvature flows coming out of generic cones of small entropy in low dimensions.
In this paper we prove a local curvature estimate for the $κ$-LYZ flow over Kähler manifolds introduced in [FGP1] and [LYZ]. In particular, we generalize the long time existence of the flow.
We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.
A possible evolution of a compact hypersurface in R^n by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a consequence that no mass drop can occur along such a flow. As a further application of the techniques used above we give a new variational formulation for mean curvature flow of mean convex hypersurfaces.
In this paper, we firstly establish an Interpolating curvature invariance between the well known nonnegative and 2-non-negative curvature invariant along the Ricci flow. Then a related strong maximum principle for the $(λ_1, λ_2)$-nonnegativity is also derived along Ricci flow. Based on these, finally we obtain a rigidity property of manifolds with $(λ_1,λ_2)$-nonnegative curvature operators.
In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.
In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.
In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of Simon \cite{Simon2015}.
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