Generic regularity for minimizing hypersurfaces in dimensions 9 and 10
Otis Chodosh, Christos Mantoulidis, Felix Schulze
研究9和10维中最小化超曲面的奇异性扰动问题，通过正则性理论证明奇异点可被扰动消除。
我们证明在9和10维的环境中，最小面积超曲面的奇异性可以被扰动消除。
We prove that singularities of area minimizing hypersurfaces can be perturbed away in ambient dimensions 9 and 10.
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We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.
Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.
In this article we prove that all boundary points of a minimal oriented hypersurface in a Riemannian manifold are regular, that is, in a neighborhood of any boundary point, the minimal surface is a $\mathcal{C}^{1, \frac14}$ submanifold with boundary.
Generic regularity for minimizing hypersurfaces in dimension 11
本文研究面积极小化超曲面的一般正则性，证明11维情形下一般光滑，并在更高维中通过小扰动将奇异集维数控制为至多$n-10-\epsilon_n$。
Improved regularity for minimizing capillary hypersurfaces
本文研究了极小毛细超曲面的奇点集正则性，证明其奇点集余维数至少为4，并在毛细角接近0、π/2或π时基于与单相Bernoulli问题的联系给出改进估计。
In this paper, we show that every $8$-dimensional closed Riemmanian manifold with $C^\infty$-generic metrics admits a smooth minimal hypersurface. This generalized previous results by N. Smale and Chodosh-Liokumovich-Spolaor. Different from their local perturbation techniques, our construction is based on a global perturbation argument in and a novel geometric invariant which counts singular points with suitable weights.
We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing.
We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics there exists a minimal hypersurface with at most one singular point. This extends previous work on generic regularity that only dealt with area-minimizing hypersurfaces. These results are a consequence of a more general estimate for a one-parameter min-max minimal hypersurface $Σ\subset (M,g)$ (valid in any dimension): $$\mathcal H^{0} (\mathcal{S}_{nm}(Σ)) +{\r…
On Simon's area-minimizing hypersurfaces with fractal singular sets
本文验证了Leon Simon构造的稳定极小超曲面为面积最小化，从而解决了具有分形奇异集的面积最小化超曲面的存在性问题。
A four dimensional conformally invariant energy is studied. This energy generalises the well known two-dimensional Willmore energy. Although not positive definite, it includes minimal hypersurfaces as critical points. We compute its first variation and by applying the Noether theorem to the invariances, we derive some conservation laws which are satisfied by its critical points and with good analytical dispositions. In particular, we show that its critical points are smooth. We also investigate other possible four dimensional generalisations of the Willmore energy, and give strong evidence tha…
Locally stable minimal hypersurface could have singularities in dimension $\geq 7$ in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this paper, we present different aspects of how these singularities may affect the local behavior of minimal hypersurfaces. First, given a non-degenerate minimal hypersurface with strictly stable and strictly minimizing tangent cone at each singular point, under any small perturbation of the metric, we show the existence of a nearby minimal hypersurface under new metric. For a generic choice of perturbation direction, we …
Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space
研究了双曲空间中渐近边界为无穷远处球面中闭余维1子流形的常中曲率超曲面。我们认为CMC超曲面是极小超曲面的推广。我们将极小超曲面的面积极小化、凸包性质、交换圆技巧等概念自然地推广到CMC超曲面的情形。我们还给出了双曲空间中CMC超曲面的一般唯一性结果。
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.
Stable constant mean curvature hypersurfaces are area minimizing in small L^1 neighborhoods
证明了在维数小于或等于7的光滑流形中，严格稳定的常平均曲率超曲面在小的L^1邻域中对固定体积唯一地同调面积极小化。
Area-minimizing Hypersurfaces in Singular Ambient Manifolds
本文研究了正则部分切锥具有非负数量曲率的奇异环境流形中的面积极小超曲面，证明了其奇异集余维数至少为3，并通过实例证明该正则性结果是尖锐的。
Warped area-minimizing hypersurface and warped product metric
本文研究扭曲面积最小超曲面，证明在谱Ricci或谱标量曲率下界条件下，度量可局部分解为扭曲积度量。
Existence and regularity of mean curvature flow with transport term in higher dimensions
给定索伯列夫空间中的初始$C^1$超曲面和依赖于时间的向量场，证明了从给定超曲面出发，以等于中曲率加上给定向量场的速度运动的超曲面族的时间整体存在性.我们证明了超曲面在短时间内是$C^1$，甚至在一些奇点出现后，几乎在远离高重数区域的任何地方都是$C^1$。
$C^{1,α}$-Regularity of energy minimizing maps from a 2-dimentional domain into a Finsler space
证明了从二维黎曼流形到具有Finsler结构$F(u,X)$的Finsler空间$(\R^n, F)$的能量极小化映射的$C^{1,α}$ -正则性.
The regularity of harmonic maps into spheres and applications to Bernstein problems
我们证明了正则性，并推导出先验估计的（弱）调和映射从黎曼流形到欧氏球面的假设下，图像避免了一些邻居的半赤道。证明结合了严格凸函数的构造和拟线性椭圆方程组的正则性理论。将这些结果应用于极小超曲面的球面和欧氏Bernstein问题，得到了球面中的紧致极小超曲面或欧几里得空间中的完备极小超曲面平凡的新条件.
On isolated singularities and generic regularity of min-max CMC hypersurfaces
本文证明正里奇曲率流形中Allen-Cahn极小极大常平均曲率超曲面在孤立奇点处局部极小且切锥面积极小，并通过手术构造一般8维流形上的闭嵌入光滑常平均曲率超曲面。
Foliation of area-minimizing hypersurfaces in asymptotically flat manifolds of higher dimension
研究在高维渐近扁平流形中面积最小超曲面的分解及其全局行为
Regularity of minimal hypersurfaces with a common free boundary
设$N$是一个黎曼流形，考虑三个或三个以上的有边$M_k$的$C^{1,μ}$超曲面在具有公共边界$Γ$的$N$中的平稳并.我们表明，如果$N$是光滑的，那么$Γ$是光滑的，每个$M_k$是光滑的$Γ$（真正的解析的情况下，$N$是真正的解析）。因此，我们加强了Wickramasekera的结果，得出结论，在更强的假设下，$V$是一个平稳的，稳定的，积分$n$ -变倍的$(n+1)$ -维，光滑（实解析）黎曼流形，使得$\|V\|$的支集在任何地方都不是三个或更多光滑流形的并（实解析）超曲面与边界沿公共边界相交，如果$n = 6$，则$V$的奇异集是空的，如果$n = 7$，则$V$是离散的，并且如果$n \geq 8$，则具有至多$n-7$的Hausdorff维数。
Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds
本文针对高维黎曼流形中的体积极小自由边界超曲面，建立了基于数量曲率和边界第二基本形式的体积估计，并在等式情形下证明了流形的刚性。
Regularity of volume-minimizing flows on 3-manifolds
在这篇文章中，我们证明了，对于任何紧的3-流形，有一个$C^{1}$体积最小的一维叶理。更一般地说，我们证明了存在的质量最小化的可求长部分的球面丛没有孤立的“极点”的基础流形。这个相同的分析被用来表明，由于Sharon Pedersen，$S^{2n+1}$的单位切丛的潜在体积最小化的可求正截面（可求正叶理）的例子实际上不是体积最小化。
Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicity
本文推广了Allard边界正则性定理，研究具有光滑边界且边界取任意重数的面积最小化流，并由此得出重要结构性结论。
Regularity and compactness theory of stable free-boundary minimal hypersurfaces
本文建立了稳定自由边界极小超曲面的Schoen-Simon正则性与紧性理论，通过倾斜函数的ε-正则性和De Giorgi迭代证明层片定理，并推广了自由边界Almgren-Pitts存在性理论到任意维数。
A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2
证明了具有弱接近重数为2的超平面的局部有限余维数为2的Hausdorff测度奇异集的开球的稳定极小超曲面在内部是2值C^{1，alpha}图.应用包括一类浸入稳定极小超曲面的紧性定理和这类超曲面在低维下的点态曲率估计也进行了讨论。
We show that if an asymptotically flat manifold with horizon boundary admits a global static potential, then the static potential must be zero on the boundary. We also show that if an asymptotically flat manifold with horizon boundary admits an unbounded static potential in the exterior region, then the manifold must contain a complete non-compact area minimizing hypersurface. Some results related to the Riemannian positive mass theorem and Bartnik's quasi-local mass are obtained.
Existence and regularity of min-max anisotropic minimal hypersurfaces
本文利用 min-max 构造在维数至少为 3 的闭光滑 Riemannian 流形中找到了奇异集余维数为 2 且 Hausdorff 测度为零的各向异性极小超曲面，并通过获得密度比的均匀上界证实了 Allard 在 1983 年提出的猜想。
Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
研究模2同调中面积最小化子流形的光滑性，证明其一般非光滑（除特定情况），解决怀特猜想，还建立奇异集豪斯多夫维数下界，并证明$\rpt$维罗内塞极小嵌入上的锥是模2面积最小化的。
Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth Riemannian manifold of dimension $m+1$. We prove that its interior singular set is, up to a relatively closed set of dimension at most $m-2$, a $C^{1,α}$ submanifold of dimension $m-1$ at which, locally, $N\leq p$ regular sheets of the current join transversally, each sheet counted with a positive multiplicity $k_i$ so that $\sum_i k_i = p$. This completes the analysis of the structure of the singular set of area minimizing hypersurfaces modulo $p$, initiated by J. Taylor for $m=2$ and $p=3$ and extended by the authors…
Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture
该研究证明4到7维渐近平坦流形可由渐近于坐标超平面的面积极小超曲面叶状化，借此证明对应正质量非负数量曲率流形中坐标柱内自由边界解的漂移性，引入全局极小超曲面概念并验证Schoen猜想的一个版本。
Minkowski content estimates for generic area minimizing hypersurfaces
该研究通过扰动方法证明了面积最小超曲面的奇异性集的Minkowski维数低于n-9。
Regularity of branched stable minimal immersed hypersurfaces
该研究建立了分支稳定极小浸入超曲面非分支奇异集的Hausdorff维数精确界，构造了达到该界的例子，并以广义Schoen不等式和分支分层定理为核心证明工具，推进了极小曲面正则性理论的研究。
Partial regularity at the first singular time for hypersurfaces evolving by mean curvature
本文研究了$(0, t_0)$上$\mathbb{R}^{n+1}$的某个开子集中光滑的、适当浸入的超曲面按中曲率演化.我们证明了在某些时空区域$B_R(y)\times (0, t_0)$中这些超曲面的第二基本形式的$p$ -可积性和$p\ge 2$-可积性意味着第一奇异集的$\mathcal{H}^{n+2-p}$ -测度在$B_R(y)$中为零。对于$p=2$，这是由Han和Sun独立建立的。我们的结果进一步推广了Xu，Ye和Zhao以及Le和Sesum以前对$p\ge n+2$的工作，在这种情况下奇异集被证明是空的.利用Ilmanen的一个定理，证明了当初始曲面有有限亏格时，$p=2$和$n=2\,$满足可积性条件。因此，在这种情况下，第一奇异集具有零$\mathcal{H}^2$ -测度。这是Brakke的主要正则性定理对于曲面的特殊情况的结论，但导出时不必施加面积连续性和单位密度假设。它遵循最近的工作头和Huisken和Sinestrari的流动封闭，$k$凸超曲面，即超曲面的总和最小的$k$主曲率是积极的，我们的可积性准则举行与指数$p=n+3-k-α$所有小$α>0$只要$1\le k\le n-1$。因此，这种解决方案的第一个奇异集是最$(k-1)$维，这是一个最佳的估计，鉴于一些明确的例子。
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $Σ\simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and the outgoing null hypersurface $\mathcal{H}$ emanating from $\partial Σ$, the time of existence of a solution to the Einstein vacuum equations is controlled by low regularity bounds on the initial data at the level of curvature in $L^2$. The proof uses the bounded $L^2$ curvature theorem by Klainerman, Szeftel and Rodnianski, the extension procedure for the constrai…
In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in scalar curvature geometry saying that any (n-1)-dim. homology class of a compact manifold M^n with positive scalar curvature can be represented by a smooth (!) compact hypersurface that admits a positive scalar curvature metric.
Interior regularity of area minimizing currents within a $C^{2,α}$-submanifold
本文证明了在 $C^{2,α}$ 类嵌入子流形内，面积极小整流的内部奇异集的 Hausdorff 维数不超过 $m-2$，确立了任意余维数下该流的内部正则性理论。
In this article we extend to generic $p$-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $p=2$. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done by Cheeger and Naber in 2013. Then, adapting the work of Naber and Valtorta, we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is $k$…
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative R…
Quantitative stratification and higher regularity for biharmonic maps
在本文中，我们证明了定量的正则性结果的平稳和最小化外部双调和映射。作为一个应用程序，我们确定尖锐的，尺寸独立的$L^p$界$\nabla^k f$，不需要一个小的能量假设。特别地，对于所有$1\le p<5/4$，每个极小化双调和映射都在$W^{4,p}$中。此外，对于最小化双调和映射从$Ω\subset \mathbb{R}^5$，我们确定了一个统一的界限的数量的奇点在一个紧集。最后，使用降维参数，我们将这些结果扩展到最小化和平稳的双调和映射到特殊的目标。
Regularity of area minimizing currents III: blow-up
这是最后一个系列的三篇论文中，我们给出了一个新的，较短的证明略有改进的版本Almgren的部分正则性面积最小化电流在黎曼流形。在这里，我们进行爆破分析推导出的区域最小化电流的规律性，从Dir-minimizing多值函数。
Let $\mathcal{H}$ denote the future outgoing null hypersurface emanating from a spacelike 2-sphere $S$ in a vacuum spacetime $(\mathcal{M},\mathbf{g})$. In this paper we study the so-called canonical foliation on $\mathcal{H}$ introduced by Klainerman and Nicolò and show that the corresponding geometry is controlled locally only in terms of the initial geometry on $S$ and the $L^2$ curvature flux through $\mathcal{H}$. In particular, we show that the ingoing and outgoing null expansions $\mathrm{tr} χ$ and $\mathrm{tr} \underlineχ$ are both locally uniformly bounded. The proof of our estimates…
In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $Σ$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of $Σ$. Furthermore, if $Σ$ saturates the respective upper bound and $M$ has nonnegative Ricci curvature, then $Σ$ is isometric to $\mathbb{S}^4$ up to scaling and $M$ splits in a neighborhood of $Σ$. Also, we obtain a rigidity result for the Riemannian…
Regularity of minimizing $p$-harmonic maps from $B^3$ to $\mathbb{S}^3$
本文证明从三维球到三维球面的极小p-调和映射在p>1.44时具有局部C^{1,α}正则性，填补了先前正则性区间的空白，并覆盖所有p≥2及次二次范围。
Regularity theory for $2$-dimensional almost minimal currents III: blowup
我们分析了$2$维积分电流在奇点在适当意义下几乎最小化的渐近行为。我们的分析是第二部分的论点，它显示了离散的奇异集为以下三类$2$维电流：面积最小化黎曼流形，半校准和球形截面的$3$维面积最小化锥。
Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - characterized by topological and analytic properties of tangent maps - is a topological manifold. In the special case of maps into the sphere ${\mathbb S}^2$, we conclude that the whole top-dimensional part of the singular set is a manifold - this generalizes a similar result in two-d…
We show that in an $8$-dimensional closed Riemmanian manifold with $C^\infty$-generic metrics, every minimal hypersurface is smooth and nondegenerate. This confirms a full generic regularity conjecture of minimal hypersurfaces in dimension eight. This also enables us to generalize many generic geometric properties of (Almgren-Pitts) min-max minimal hypersurfaces, previously only known in low dimensions, to dimension eight.
A 7-dimensional area-minimizing embedded hypersurface $M$ will in general have a discrete singular set. The same is true if $M$ is stable, or has bounded index, provided $H^6(sing M) = 0$. We show that if $M_i$ are a sequence of such minimal hypersurfaces which are minimizing, stable, or have bounded index, then $M_i$ can limit to a singular $M$ with only very controlled geometry, topology, and singular set. We show one can always "parameterize" a subsequence $i'$ with controlled bi-Lipschitz maps $ϕ_{i'}$ taking $ϕ_{i'}(M_{1'}) = M_{i'}$. As a consequence, we prove the space of smooth, closed…
We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (e.g. integrability). Moreover, if the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new $\varepsilon$-regularity result for almost …
Partial regularity of mass-minimizing Cartesian currents
设B是紧黎曼流形M上的具有紧纤维F的纤维丛。在全空间B上存在一个与M上的度量一致的自然黎曼度量。关于该度量，可整流部分s：M--> B的体积是作为B中的可整流n电流的图像s（M）的质量。定理：对于B的任意同调类截面，存在一个质量最小化的笛卡尔流T，它表示M的开稠密子集上的C^1截面的图的同调类。
We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called S-structure which reveals some unexpected geometric and analytic properties of the hypersurface and its singularity set. In this paper, this is used to prove the existence of hyperbolic unfoldings: canonical conformal deformations of the regular part of these hypersurfaces into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorp…
We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
Lightlike hypersurfaces and time-minimizing geodesics in cone structures
本文将洛伦兹几何中的类光超曲面概念推广到无度规的锥结构，证明其具有唯一锥测地线叶状结构，并借此在全局双曲时空中获得锥测地线的时间最小化性质。
Area minimizing hypersurfaces modulo $p$: a geometric free-boundary problem
本文研究完备 Riemann 流形中模 $p$ 面积极小流的局部结构，证明其向具有脊的切锥唯一收敛并衰减，从而给出奇数模数及偶数模数相应条件下的自由边界分解。
Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality
该研究针对取值于3维球面的3维单位球上的极小$p$-调和映射，结合经典方法与新工具拓展了已知正则性的$p$值区间，还证明了二维区域下的精确Kato不等式。
Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below
本文利用Cheeger-Colding理论，研究Ricci曲率下有界流形中面积极小超曲面在Gromov-Hausdorff收敛下的体积连续性，并通过blowing up论证给出极限超曲面奇异集的精确维数估计。
A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces
证明了标题中的两个定理，解决了奇异极小超曲面局部理论中两个长期存在的问题。任何一个结果的尖锐性都与其关于允许奇异集的大小的假设有关。这两个定理的证明在很大程度上依赖于作者最近的稳定极小超曲面的正则性和紧性理论，以及Ilmanen，Simon和Solomon-White的早期工作。
We introduce a $\mathbb{Z}$--coefficient version of Guth's macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homology classes. We give examples to show that a very positive lower bound on macroscopic scalar curvature does not necessarily imply an upper bound on the areas of minimizing hypersurfaces.
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