Massimiliano Morini, Marcello Ponsiglione, Emanuele Spadaro
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.
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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.
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).
We show stability of spherical caps (SCs) lying on a flat surface, where the motion is governed by the volume-preserving Mean Curvature Flow (MCF). Moreover, we introduce a dynamic boundary condition that models a line tension effect on the boundary. The proof is based on the generalized principle of linearized stability
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.
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.
We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^{2n})$ suitably converging we construct {for $n\geq 3$} a sequence $(u_k)$ of radially symmetric solutions to the equation $${(-Δ)^n u_k=V_k e^{2n u_k} \quad \text{in }\mathbb{R}^{2n},}$$ with $u_k$ blowing up at the origin \emph{and} on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the $4$-dimensional case studied by F. Robert (J. Diff. Eq. 2006).
We consider a mass conserved Allen-Cahn equation $u_t=Δu+ \e^{-2} (f(u)-\eλ(t))$ in a bounded domain with no flux boundary condition, where $\eλ(t)$ is the average of $f(u(\cdot,t))$ and $-f$ is the derivative of a double equal well potential. Given a smooth hypersurface $γ_0$ contained in the domain, we show that the solution $u^\e$ with appropriate initial data approaches, as $\e\searrow0$, to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from $γ_0$.
During his experiments W. D. Ristenpart made a very remarkable discovery. If two oppositely charged droplets of fluid are close enough, at first they attract each other and touch eventually. Surprisingly after that the droplets are repelled from each other, if the initial strength of the charges is high enough. Otherwise they coalesce and form a big drop, as one might expect. We present a theoretical model for this observation, using mean curvature flow. With the help of appropriate barriers for the flow we can predict the observed droplet behaviour. This shows that, contrary to general belief…
We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by volume-preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non-local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a Hanzawa transformation to write the fl…
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…
We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th 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.
Diffusion-aggregation equations and volume-preserving mean curvature flows
本文针对椭圆-抛物型PKS模型，在二维和三维中首次无条件证明其解收敛到保体积平均曲率流，通过Brakke型公式和带非局部项的Allen-Cahn方程重释，避免了相分离无能量损失的假设。
Existence of weak solution to volume preserving mean curvature flow in higher dimensions
本文通过Allen--Cahn方程构造了高维体积保持平均曲率流的全局弱解，并证明了在初始数据周长接近球体时解的收敛性。
A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow
本文提出基于非线性二阶椭圆算子的Heisenberg群水平拟凸函数PDE方法，给出粘性下解的充分必要条件，并证明水平曲率流在演化中保持h-凸性。
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.
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.
Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions
该研究针对平面体积约束下的各向异性曲率驱动集合运动问题，采用Cahn-Hoffman映射参数化边界，证明保面积各向异性平流指数收敛到等面积Wulff形不交并，还得到收敛轮廓与初始数据距离的一致界。
We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).
Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles
本研究结合约束最小化运动格式与离散到连续分析，研究二维晶格晶体的保面积平均曲率型运动，刻画了连续与离散模型的控制方程，并证明时空尺度同速率趋于零时离散到连续演化由不可退化为ODE的微分包含系统描述。
Convergence of the volume preserving fractional mean curvature flow for convex sets
本文证明从凸集出发的保体积分数阶平均曲率流不产生奇点，并利用正则性提升论证，结合已有结果，得出该流以指数速度收敛到球。
Existence of weak solutions to volume-preserving mean curvature flow with obstacles
通过相场方法证明了所有维度下带障碍的体积保持平均曲率流的全局弱解存在性，基于Allen-Cahn方程与乘子的收敛，并处理了障碍引起的空间依赖强迫项。
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…
Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions
本文研究了体积保持均曲率流在三维和二维中的最终正则性，证明了流在有限时间后变得光滑，并以经典意义满足方程，且在所有C^k范数下以指数速度收敛到极限球。
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。
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…
A classical result of Nitsche \cite{Nit57} about the behaviour of the solutions to the Liouville equation $Δu=4 e^{2u}$ near isolated singularities is generalized to solutions of the Gaussian curvature equation $Δu=- κ(z) e^{2u}$ where $κ$ is a negative Hölder continuous function. As an application a higher--order version of the Yau--Ahlfors--Schwarz lemma for complete conformal Riemannian metrics is obtained.
We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modelled by an interaction potential. We consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution's support. The analytical results are supported by numerical simulations for…
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。
Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation
本文研究了椭圆-抛物PKS模型，证明在非线性扩散下发生相分离现象，且界面运动渐近地由保体积平均曲率流描述。
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 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.
Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds
本文研究了数据科学背景下加权流形上的多相体积约束MBO阈值化方案，证明了其在光滑闭流形上向加权保体积平均曲率流的收敛性，并提出了新的De Giorgi解概念。
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。
Area-preserving curvature flow for open curves with general contact angles on skew lines
本文研究斜线上具有一般接触角的平面开曲线的保面积曲率流，证明了全局存在唯一性及指数收敛到等面积圆弧。
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。
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。
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…
We study the asymptotic behaviour of the $p$-capacitary potential and of the weak Inverse Mean Curvature Flow of a bounded set along the ends of an Asymptotically Conical Riemannian manifolds with asymptotically nonnegative Ricci curvature.
We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such ma…
In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provided the radius satisfies a certain condition. Here we consider flows where there is a critical radius such that the cylinders are only stable on one side of the critical value, and the weight function is a linear combination of the elementary symmetric functions, the reason for s…
Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space
研究双曲空间中保体积非齐次高斯曲率流，证明解在所有正时间存在且指数收敛到测地球，关键在于高斯曲率振荡衰减与双曲曲率测度论结合推导出 Hausdorff 收敛。
The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere
本研究提出球面中horo-凸超曲面的完全非线性曲率流，可保持任意指定球面quermassintegral，证明了horo-凸性保持、长时间存在及指数收敛到对应测地球面。
Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow
本文建立更一般的定量 Alexandrov 定理，并据此证明二维、三维中由最小化运动格式构造的保体积平均曲率流平坦流渐近收敛为等大互不相交球的平移并集。
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such a technique. We will consider surfaces with boundary contained within two parallel planes such that the surface meets these planes orthogonally. With this set up the stationary solution is a cylinder. We will find that for initial surfaces that are sufficiently close to a cylinde…
Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness
本文提出了一种新的 varifold 解概念，用于双相体积保持均 curvature 流，证明了其存在性和弱强唯一性。
Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces
本文研究渐近平坦3-流形中曲面的保体积平均曲率流，证明在正ADM质量和初始曲面为圆时，流长期存在并光滑收敛到稳定CMC曲面，推广了Huisken-Yau的结果，并提供构造CMC叶状结构的新方法。
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。
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