Brian White
We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow.
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A Curvature Varifold Whose Weak Second Fundamental Form is Not Preserved Under Decompositions
本文构造了一个曲率varifold，证明其无法分解为曲率varifold分量，揭示了弱第二基本形式在分解下的不保持性。
A Corrected Proof of the Graphical Representation of a Class of Curvature Varifolds by $C^{1,α}$ Multiple Valued Functions
本文通过反例指出 Hutchinson 原证明的缺陷，提供替代证明并引入变分图嵌套分解方法，同时证明零第二基本形式曲率变分的结构定理。
A sharp lower bound on the mean curvature integral with critical power for integral varifolds
本文为数学专著目录，涵盖弱可微函数、变分上的偏微分方程及广义Gauss映射，重点探讨积分变分平均曲率积分临界幂次的精确下界。
Integral Brakke approximate equality for an approximate mean curvature flow
本文通过体积变差测度离散化，推导出平均曲率流的积分Brakke近似等式，以研究其一致性。
We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and the boundedness of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.
We prove that the support of an $ m $ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $ \mathscr{H}^{m} $ almost everywhere by a countable union of $m$ dimensional submanifolds of class $ \mathcal{C}^{2} $. We obtain this result using the notion of curvature of arbitrary closed sets originally developed in stochastic geometry and extending to our geometric setting techniques developed by Trudinger in the theory of viscosity solutions of PDE's.
Global regularity of integral 2-varifolds with square integrable mean curvature
通过分析Hausdorff密度和Willmore能量阈值，证明了具有临界可积平均曲率的积分2-曲面由光滑曲面的W^{2,2}共形浸入诱导，并建立了曲率曲面与参数方法的等价性。
The optimal isoperimetric inequality for varifolds
本文证明了Brendle等周不等式对有限质量与全变差的varifolds成立，允许奇异部分，仅需可积平均曲率，常数在余维数一、二时最优。
Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions
本文证明了由环境函数规定平均曲率的毛细超曲面的紧致性定理，推广了Schätzle和Bellettini的结果，并扩展了曲率varifold定义至有向积分varifold，同时讨论了边界平均曲率规定情形。
We prove that every stationary polyhedral varifold minimizes area in the following senses: (1) its area cannot be decreased by a one-to-one Lipschitz ambient deformation that coincides with the identity outside of a compact set, and (2) it is the varifold associated to a mass-minimizing flat chain with coefficients in a certain metric abelian group. NOTE: After this paper was posted, I learned that (1) and (2) were already proved by Choe and Morgan, respectively. Thus this paper is an exposition of their results.
Curvature Varifolds with Orthogonal Boundary
本文研究带正交边界的曲率varifolds，给出面积估计、零曲率分类，并证明任意区域中最小化L²曲率的正交2-varifold的存在性。
A barrier principle at infinity for varifolds with bounded mean curvature
本文在黎曼流形开区域内研究任意余维、有界平均曲率的变分曲面，在允许边界奇异性的温和曲率条件下证明无穷远障碍原理，表明曲面到区域边界的距离在曲面边界上取得，并由此得到具有独立意义的尖锐无穷远极大值原理。
We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] -…
De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach
提出平均曲率流和体保持平均曲率流全局弱解的存在性证明，首次证明最小化运动方案无条件收敛到De Giorgi变分形式解，论证纯变分且不依赖比较原理。
In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere o…
Critical Allard regularity: pointwise tilt-excess estimates
本文为具有临界广义平均曲率的几乎整数 varifold 建立了二次倾斜超额的 VMO 型逐点估计，并推广了 $Q$-值 Lipschitz 逼近与 Sobolev-Poincaré 估计，进而推导出 Reifenberg 型参数化。
Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flow
本研究基于变分体理论提出一般变分体的近似平均曲率流构造方法，证明其收敛性与多项性质，并在可求长假设下得到极限为时空Brakke流的结论。
We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is $g ν$, for a given continuous positive ambient function $g$, and where $ν$ denotes the inner normal. It is well-known that taking limits in the sense of varifolds within this class is not possible in general, due to the appearence of "hidden boundaries", that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that $g$ does not prescribe the mean curvature in the limit. As a special instance of a…
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。
In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural conditions. Here we generalize this theory to the setting where the mean curvature (of the regular part of the varifold) is prescribed by a given $C^{1, α}$ ambient function $g$ and the ambient space is a general $(n+1)$-dimensional Riemannian manifold; we give general conditions that imply that the support of the varifold, away from a possible singular set of dimension $\leq n-7,$ is the image of a $C^{2}$…
Global weak varifold solutions to Navier-Stokes/Mean Curvature Flow system
本文针对Navier-Stokes/平均曲率流系统，提出基于De Giorgi能量耗散原理的varifold弱解概念，首次证明二维和三维中无条件全局时间存在性，并利用相对熵方法证明经典解在该弱解类中的唯一性。
In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywhere with respect to the varifold.
In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, which allows to extend the classical Coarea formula for the Gauss map of smooth varieties, and to introduce for all integral varifolds of bounded mean curvature a natural defin…
The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts identities for certain compositions with smooth functions. In this class the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincaré type embeddings, embeddings into spaces of continuous and sometimes Hölder continuous functions, pointwise differentiability results both of approximate a…
We survey - by means of 20 examples - the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one considers, for instance, existence or regularity of stationary (or, stable) surfaces of dimension at least three, or the limiting behaviour of sequences of smooth submanifolds under area and mean curvature bounds.
Spacelike Brakke flows with boundary in pseudo-Euclidean space
研究伪欧几里得空间中带边界的类空平均曲率流，通过建立弱形式、引入相关向量，证明闭包和紧致性定理等，采用椭圆正则化程序证明存在性，为研究奇点外的弱类空平均曲率流提供变差设置。
Quantization of the Energy for the inhomogeneous Allen-Cahn mean curvature
本文研究非齐次 Allen-Cahn 相变问题，证明了相场极限下 Dirichlet 与 Potential 能量均分，且总能量 varifold 收敛于具有 $L^{q_0}$ 平均曲率的整数可整 varifold，是 Allard 定理的扩散版本。
Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature
本文研究了具有有界各向异性均曲率的变分的法向性和局部性，扩展了Brakke、Schätzle和Ambrosio-Masnou在欧几里得空间中的经典定理。
Optimal rigidity estimates for varifolds almost minimizing the Willmore energy
本文研究几乎最小化 Willmore 能量的积分 2-varifold 的刚性，证明其在小扰动下定量 W^{2,2} 接近标准球面，并在三维情形确定尖锐常数 δ_0^2=2π。
Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature
该研究证明了具有有界各向异性均曲率的变分在几乎所有点上具有正则性，结合Allard定理进一步表明其在密度为1的点上具有更高级的正则性。
We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally…
This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4πmust be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.
We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of the $C^{1}$-embedded part of the varifold (which is non-empty by the Allard regularity theory) is stationarity and the $C^{2}$-immersed part of it is stable with respect to the area functional for volume preserving deformations, its support, except possibly on a closed set of codimension $7$, is an immersed constant-mean-curvature (cmc) hypersurface of class $C^{2}$ that can fail to be embedded only…
We prove a Poincaré, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature bounded above. Our techniques allow us to consider Riemannian manifolds $(M^n,g)$ with $g$ of class $C^2$ or more regular, avoiding the use of Nash's isometric embedding theorem. Our analysis permits to do some quite important fragments of geometric measure theory also for those Riemannian manifolds carrying a $C^2$ metric $g$, that is not $C^{k+α}$ with $k+α>2…
We develop a suitable generalization of Almgren's theory of varifolds in a lorentzian setting, focusing on area, first variation, rectifiability, compactness and closure issues. Motivated by the asymptotic behaviour of the scaled hyperbolic Ginzburg-Landau equations, and by the presence of singularities in lorentzian minimal surfaces, we introduce, within the varifold class, various notions of generalized minimal timelike submanifolds of arbitrary codimension in flat Minkowski spacetime, which are global in character and admit conserved quantities, such as relativistic energy and momentum. In …
Regularity of minimal surfaces with capillary boundary conditions
本文证明了黎曼流中具有毛细边界条件的变分折叠的ε-正则性定理，建立了均匀的第一变分控制，并推导出在密度严格小于1的边界一般点处的正则性。
If one considers an integral varifold $I^m\subseteq M$ with bounded mean curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is }k+1\text{-symmetric}\}$ is the standard stratification of the singular set, then it is well known that $\dim S^k\leq k$. In complete generality nothing else is known about the singular sets $S^k(I)$. In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum $S^k(I)$ is $k$-rectifiable. In fact, we prove for $k$-a.e. point $x\in S^k$ that there exists a unique $k$-plan…
Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities
本文通过最优质量输运，为积分varifolds建立了等周和Michael-Simon-Sobolev不等式，推广了Brendle-Eichmair结果，并推广了Alexandrov定理，常数在余维2时最优。
We prove that $m$-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p>m$, are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a $C^2$ neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic c…
This paper presents a conditional convergence result of solutions to the Allen--Cahn equation with arbitrary potentials to a De Giorgi type $ \mathrm{BV} $-solution to multiphase mean curvature flow. Moreover we show that De Giorgi type $\mathrm{BV} $-solutions are De Giorgi type varifold solutions, and thus our solution is unique in a weak-strong sense.
The Structure of Stable Codimension One Integral Varifolds near Classical Cones of Density 5/2
本研究针对密度为5/2的经典锥附近的稳定余维一积分变分，发展精细爆破方法证明多值$C^{1,α}$正则性定理，完成密度小于3区域奇异集的分析，填补了高重数非平坦锥相关研究的空白。
Allard-Type Regularity for Varifolds with Prescribed Contact Angle
本文证明了在给定域边界上具有 prescribed contact angle 的积分 varifold，在 tangent cone 为 density one 的 half-hyperplane 条件下，其支撑集在边界点附近是 $C^{1,γ}$ 超曲面。
Uniqueness of Finite-Time Varifold Limits for the Möbius-Invariant Willmore Flow
该研究在一致定量非脐性等条件下，证明了三维球面中莫比乌斯不变威尔莫尔流有限时间变分曲面极限的唯一性，还将结果推广至无穷时间情形，推进了几何分析中相关流的收敛性研究。
Dynamical instability of minimal surfaces at flat singular points
该研究针对具有平坦奇点的一重平稳Varifold，在适当衰减率假设下证明存在非常值Brakke流，揭示Brakke流的非唯一性，为用平均曲率流稳定性排除平坦奇点提供依据。
The anisotropic Michael-Simon inequality
本文证明了任意维数和余维数varifolds的各向异性Michael-Simon不等式，结合相关定理给出超曲面正则性，还得出varifolds的密度界与紧性性质，证明依赖投影方法及消失质量猜想的解决。
Convex hulls and area minimizing currents in asymptotically locally hyperbolic manifolds
该研究在无全局曲率假设的共形紧化渐近局部双曲流形中，通过构造共形无穷远处的障碍凸包，推广Anderson的双曲空间定理，求解任意余维数整电流的定向渐近普拉托问题。
A priori bounds for geodesic diameter. Part III. A Sobolev-Poincaré inequality and applications to a variety of geometric variational problems
本文基于新型 Sobolev-Poincaré 不等式，建立了用曲面及边界平均曲率表示任意维广义紧致带边曲面测地直径的有限上界，并适用于多种含奇点的 varifold 与 Plateau 问题解。
We consider a short time existence problem motivated by a conjecture of Joyce. Specifically we prove that given any compact Lagrangian $L\subset \mathbb{C}^n$ with a finite number of singularities, each asymptotic to a pair of non-area-minimising, transversally intersecting Lagrangian planes, there is a smooth Lagrangian mean curvature flow existing for some positive time, that attains $L$ as $t \searrow 0$ as varifolds, and smoothly locally away from the singularities.
We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian cycles (possibly with differing phases). We use the flow of cycles to obtain the fact that special lagrangian cycles generate the part of the lagrangian homology which lies in the image of the Hurewicz homomorphism. We also establish a weak version of a conjecture of Thomas-Yau r…
On the avoidance principle for codimension 1 space-time Brakke flows
本文证明由Buet等人构造的时空Brakke流在初始varifold为有限周长区域边界并集时非平凡，并证明余维1情形下其质量测度支集与经典平均曲率流支集一致。
We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem.
The Brakke-BV structure of flat flows
该研究证明环境维数≤7时，Almgren等提出的极小运动格式的变分极限为整Brakke流，其L¹极限点（平流）为广义BV流，结果无条件且适用于无能量收敛、存在粗化的情况。
We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asy…
We prove a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds.
A Structure Theory for Stable Codimension 1 Integral Varifolds with Applications to Area Minimising Hypersurfaces mod p
本文建立无低密度经典奇点的稳定余维1平稳积分varifold的结构理论，并用于刻画模p面积极小超曲面在平面切锥点附近的唯一性与广义$C^{1,\alpha}$分支结构。
We construct real analytic flat Moebius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature.
In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive mass, the foliation of stable spheres of constant mean curvature is unique.
We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.
Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic structure of the piecewise flat manifold and a choice of dual tessellation, and can be viewed as the average of $n$-volume integrals. The constructions are also independent of the manifold dimension.
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