Luca Mugnai, Christian Seis, Emanuele Spadaro
In this paper, we construct global distributional solutions to the volume-preserving mean-curvature flow using a variant of the time-discrete gradient flow approach proposed independently by Almgren, Taylor and Wang (SIAM J. Control Optim. 31(2): 387- 438, 1993) and Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2): 253-271, 1995).
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In this paper we analyse the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.
We consider the volume preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long time asymptotics approach round spheres. The proofs are based on apriori estimates on the inner and outer radii of the solutions.
Stability of Spherical Caps under the Volume-Preserving Mean Curvature Flow with Line Tension
我们表明稳定的球冠（SC）躺在一个平坦的表面上，其中的运动是由体积保持中曲率流（MCF）。此外，我们引入了一个动态边界条件，模拟线张力效应的边界。证明是基于广义线性化稳定性原理
We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature flow of a periodic graph.
In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.
Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow
本文提出保体积平均曲率流的新Brakke不等式概念，用相场方法证明积分可变体全局Brakke流的存在性，扩展了相关前期研究结果。
A variational approach to the volume-preserving anisotropic mean curvature flow in 2D
本文提出一种基于最小化移动的变分算法，用于二维体积保持各向异性平均曲率流，证明经典解存在性及算法收敛于全局解。
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.
We study the evolution of star-shaped sets in volume preserving mean curvature flow. Constructed by approximate minimizing movements, our solutions preserve a strong version of star-shapedness. We also show that the solutions converges to a ball as time goes to infinity. For asymptotic behavior of the solutions we use the gradient flow structure of the problem, whereas a modified notion of viscosity solutions is introduced to study the geometric properties of the flow by moving planes method.
A quantitative Brenier decomposition shows that the deviation of a map from volume preserving is bounded by the deviation of the derivative from volume preserving. A study of the matrix nearness problem for $SL(n)$ and $Sp(2n)$ relates the estimate to incompressible deformations of elastic materials.
In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus $\mathbb T^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set in $\mathbb T^n$ in $H^3$-sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in $W^{2,5}$-sense.
We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an $m$-fold circle as time goes to infinity. For the area-preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in finite time or not, while for the length-preserving flow, it is the positivity of an energy associated with initial curve that plays such a role.
We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball. As an intermediate result we establish a quantitative Alexandrov type estimate in the fractional setting for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero.
Mass conserved Allen-Cahn equation and volume preserving mean curvature flow
考虑有界域上无通量边界条件的质量守恒Allen-Cahn方程$u_t=Δu+ \e^{-2} (f(u)-\eλ(t))$，其中$\eλ(t)$是$f(u(\cdot,t))$的平均值，$-f$是双等势阱的导数。给定一个光滑的超曲面$γ_0$包含在域中，我们表明，解决方案$u^\e$与适当的初始数据的方法，作为$\e\searrow0$，到一个极限，它只需要两个值，与跳跃发生在超曲面从体积保持中曲率流从$γ_0$。
In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai-Seis-Spadaro (2016). This method is based on the gradient flow approach due to Almgren-Taylor-Wang (1993) and Luckhaus-Strurzenhecker (1995) and my aim is to replace the volume penalization by implementing the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on the second variation condition of the…
Local Well-Posedness for Volume-Preserving Mean Curvature and Willmore Flows with Line Tension
考虑线张力对边界的影响，证明了由保体积中曲率流（MCF）驱动的与固体容器接触的演化超曲面运动解的短时存在唯一性.由于动态边界条件和接触角以及由此产生的二阶非线性偏微分方程的非局部性质，出现了困难。此外，我们证明了同样的结果为Willmore流与线张力，这将导致一个非线性偏微分方程的四阶。对于这两个流，我们将使用Hanzawa变换将流写为固定参考超曲面上的图。
Existence of flat flows for volume-preserving mean curvature flow with contact angle
本文证明了半空间中具有体积约束和接触角条件的平均曲率流存在全局弱解（平坦流），通过接触点爆破分析克服了近似解有界性等困难。
Quantitative convergence of the nonlocal Allen--Cahn equation to volume-preserving mean curvature flow
本文通过引入带切向分量的新梯度流校准概念，结合相对熵方法，证明了非局部 Allen-Cahn 方程到保体积平均曲率流的最优定量收敛速率，且无需证明 Lagrange 乘子的接近性。
We consider the flat flow solution, obtained via discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from C^{1,1}-regular set. We prove the consistency principle which states that (any) such flat flow agrees with the classical solution as long as the latter exists. In particular, the flat flow is unique and smooth up to the first singular time. We obtain the result by proving the full regularity for the discrete time approximation of the flat flow such that the regularity estimates are stable with respect to the time discretization. Our method can also be…
Volume-preserving flow by powers of the m-th mean curvature
研究了闭凸超曲面在保体积曲率流下的演化。速度由第m个中曲率的幂加上体积保持项给出，包括中曲率或高斯曲率的幂的情况。我们证明了，如果初始超曲面满足适当的拼挤条件，则解始终存在并收敛于一个圆球。
We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density upper bound for the reaction diffusion equation.
Mixed volume preserving curvature flows in hyperbolic space
考虑双曲空间中具有单调、对称、齐次一次曲率函数F的曲率流。此外，我们假设F是凹的和反凹的或凸的。对于紧致的初始超曲面，证明了其在球面上是严格凸的情形下，混合保体积曲率流的长时间存在性。此外，我们表明，这些流指数收敛在C-无穷拓扑的测地球。
Stability of the surface area preserving mean curvature flow in Euclidean space
我们证明了当初始无迹第二基本型的L^2-范数很小（但初始超曲面不一定是凸的）时，欧几里得空间中的保表面积中曲率流始终存在，并指数收敛于一个圆球.
In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The first main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms of a relative entropy, which is valid in the class of distributional solutions to volume-preserving m…
We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows - surfaces of revolution - in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additional curvature assumptions being imposed, while axial symmetry and boundary conditions are justifiable given the volume constraint. Additional results and ingredients towards the main proof include a non-cylindrical parabolic maximum principle, and a series of estimates on geometr…
Asymptotic behavior of a diffused interface volume-preserving mean curvature flow
本文研究了欧氏空间中保体积平均曲率流的扩散界面版本，证明了在所有维度及自然初始假设下，该流指数收敛至单个“扩散球”。
Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow
该研究提出半空间中带毛细边界超曲面的保体积（或保面积）幂平均曲率流，并证明任意凸初始超曲面均可全局存在且光滑收敛至球冠。
We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that any moving curve converges to a traveling wave if the moving curve starts from an embedded convex curve and remains bounded in global time.
We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.
We consider the dynamic property of the volume preserving mean curvature flow. This flow was introduced by Huisken who also proved it converges to a round sphere of the same enclosed volume if the initial hypersurface is strictly convex in Euclidean space. We study the stability of this flow in hyperbolic space. In particular, we prove that if the initial hypersurface is hyperbolically mean convex and close to an umbilical sphere in the $L^2$-sense, then the flow exists for all time and converges exponentially to an umbilical sphere.
The quermassintegral preserving mean curvature flow in the sphere
本文引入球面上凸超曲面带全局项的平均曲率流，可保持任意拟质量积分固定，证明其从严格凸初始超曲面出发全局存在且收敛到测地球面，解决了Huisken 1987年相关问题，并分类了空间形式中常曲率型方程解及孤子。
Asymptotic circularity of immortal area-preserving curvature flows
本文证明闭平面曲线保面积曲率流的永生解渐近变圆，解决 Escher--Ito 2005 年开放问题并推广至任意阶曲面扩散流，同时建立非平凡永生解的存在性定理。
The total Q-curvature, volume entropy and polynomial growth polyharmonic functions (II)
本文作为前期研究的延续，在全维度下将有限总Q曲率的完备度量刻画为正规度量，还引入新体积熵分析有限总Q曲率的完备非正规度量的几何性质，证明标量曲率有下界时其体积增长可控。
Lower volume growth estimates for Self-shrinkers of mean curvature flow
得到了中曲率流的完全非紧自收缩体的一个Calabi-Yau型较低体积增长估计，更精确地说，每个完全非紧适当浸入的自收缩体至少具有线性体积增长.
Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space
本文研究了Schwarzschild空间中保体积与保面积平均曲率流的稳定性，证明了在初始超曲面充分接近坐标球面或等周超曲面时，流全局存在并光滑或指数收敛到常平均曲率超曲面。
The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus
本文研究平面平坦环面上 Mullins-Sekerka 流与保面积曲率流的平坦流解的渐近行为，利用尖锐定量 Alexandrov 不等式证明其以指数速度收敛到圆盘、条带或其补集的有限并集。
Volume-constrained minimizers for the prescribed curvature problem in periodic media
我们建立了周期介质中具有体积约束的指定中曲率问题的紧极小解的存在性。作为结果，我们构造紧凑的近似解指定的中曲率方程。我们还表明收敛后，重新缩放的体积约束极小值对一个合适的武尔夫形状，当体积趋于无穷大。
We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.
We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topolo…
Volume growth, curvature decay, and critical metrics
我们改进了以前的结果。首先，我们证明了我们的体积增长定理的一个版本，它不需要任何假设的第一个贝蒂数。其次，我们证明了我们的局部正则性定理只需要一个较低的体积增长假设，而不是一个完整的Sobolev常数界。这些结果使我们能够削弱我们以前的体积增长和收敛定理的假设。
For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^{k}$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least …
Decay of correlations on Abelian covers of isometric extensions of volume-preserving Anosov flows
研究了封闭流形阿贝尔覆盖上保体积阿诺索夫流的各向同性扩展的关联函数的渐近衰减行为。
Volume preserving spacetime mean curvature flow and foliations of initial data sets
本文研究具有正ADM能量的渐近欧几里得初始数据集中体积保持时空平均曲率流，证明其长期存在并收敛到常数曲率极限，为CSTMC叶状结构提供新构造，应用于广义相对论中孤立系统质心的定义。
A modified phase field approximation for mean curvature flow with conservation of the volume
针对体积约束的平均曲率流，提出一种修正相场方程，相比经典非局部Allen-Cahn方程具有更好的体积保持性质。
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