Jacob Bernstein, Lu Wang
We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain sense.
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We develop a min-max theory for asymptotically conical self-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable self-expanders that are asymptotic to the same cone and bound a domain, there exists a new asymptotically conical self-expander trapped between the two.
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant cones are rotationally symmetric.
We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expanders of low entropy and for the set of mean convex self-expanders with strictly mean convex asymptotic cones. From this we deduce that the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the…
We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper. As an application we show that there is an open set in the space of cones in the three-dimensional Euclidean space for which each cone in the set has a strictly unstable self-expanding annuli asymptotic to it.
Unique continuation at infinity for conical Ricci expanders
建立了扩展梯度Ricci孤子的加权拉普拉斯算子的Carleman不等式。作为结果，证明了渐近Ricci平坦Ricci展开式在无穷远处的唯一延拓.无穷远处的障碍物是定义在相应的渐近锥面链环上的对称2-张量。
Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow
针对平均曲率流中的平均凸渐近锥形自膨胀子，论文证明了当 $n\geq 3$ 且渐近锥平均凸、弱凸时，该完备自膨胀子必为严格凸。
Stability of asymptotically conical gradient Kähler-Ricci expanders
本文证明渐近锥形梯度Kähler-Ricci扩张子在适当扰动下，规范化Kähler-Ricci流全局存在并收敛到同类的孤子度量，且若扰动在无穷远处渐近于原度量，则极限度量与原度量相同。
Existence of monotone Morse flow lines of the expander functional
针对严格不稳定的光滑渐近锥形自扩张子，构造了连接其与稳定自扩张子的单调奇异Morse流线，该流线奇异集小，证明了扩张子泛函单调Morse流线的存在性。
Degree theory for 4-dimensional asymptotically conical gradient expanding solitons
本文为4维渐近锥形梯度扩张孤子建立新度理论，证明对任意非负标量曲率锥存在此类孤子，并引入扩张子度不变量，揭示梯度与非梯度孤子的连通性关系。
Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders
本文证明了渐近于同一锥的高余维自收缩子唯一且以单位重数收敛，并给出自扩张子在此条件下像重合的唯一性结果。
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 use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for self-expanders of the mean curvature flow that flow from a cone.
Deformations of Asymptotically Conical Special Lagrangian Submanifolds
麦克莱恩研究了变形紧凑的特殊拉格朗日子流形，特别表明，他们来在模空间的尺寸只取决于拓扑的子流形。在这篇文章中，我们研究了类似的问题，非紧的，“渐近圆锥”SL子流形，关于各种“边界条件在无穷远”。
$G_2$-Laplacian flow on complete asymptotically conical manifolds
本文证明了完备流形上闭的渐近锥形$G_2$-结构的$G_2$-拉普拉斯流的短时间存在性与闭解中的唯一性，给出了依赖初始几何的存在时间下界。
An Isomorphism Theorem for Some Linear Elliptic Differential Operators on Quasi-Asymptotically Conical Manifolds
该研究针对拟渐近锥形流形，探究了一类带无界$b$-向量场的线性椭圆微分算子，为相关几何分析问题提供算子层面的理论支撑。
Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for…
Fredholm theory for elliptic operators on quasi-asymptotically conical spaces
研究了拟渐近锥（QAC）空间类上广义Laplace型算子${\mathcal L} = \nabla^* \nabla + {\mathcal R}$的映射性质，给出了Joyce所考虑的QALE流形的黎曼推广.我们的主要结果给出的条件下，这些运营商是Fredholm之间的某些加权Sobolev或加权Hölder空间。这些都是在渐近圆锥（或渐近欧几里德）设置众所周知的定理的推广，也锐化和扩展相应的定理乔伊斯。这里的方法是基于热核估计起源于旧的想法莫泽和纳什，作为进一步发展的格里戈尔'扬和Saloff-Coste。正如乔伊斯的工作所证明的那样，这里的QAC空间包含了许多引力瞬子的例子，这项工作的动机是对具有特殊完整性的流形的各种应用。
Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
该研究证明在闭合流形上存在一个里齐流解，其在有限时间形成基于渐近圆锥收缩子流形的奇异性和模型。
For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all non-flat two-dimensional self-shrinkers. This confirms a conjecture of Colding-Ilmanen-Minicozzi-White in dimension two.
If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow.
Curvature pinching of asymptotically conical gradient expanding Ricci solitons
本文研究了完整非紧致渐近锥形梯度扩展里奇溶剂的曲率捏紧现象，建立了若干Hamilton-Ivey型曲率捏紧估计，包括三维梯度里奇扩展体的正截面曲率结果。
Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends
设$C\subset\mathbb{R}^{n+1}$是一个顶点位于原点的正则锥。本文证明了$\mathbb{R}^{n+1}$中光滑适当嵌入的自缩端的唯一性，即渐近于$C$ .作为应用，我们证明了不是每一个顶点在原点的正则锥都有光滑的完全恰当嵌入自收缩渐近。
We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G_2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G_2-cone, then their G_2-structures are equivalent, and in particular, the two solitons are isometric. The proof extends Kotschwar and Wang's argument for uniqueness of AC gradient shrinking Ricci solitons. We additionally show that the symmetries of the G_2-structure of an AC shrinker end are inherited from its asymptotic cone; under a mild assumption on the fundamental g…
Rigidity of asymptotically conical shrinking gradient Ricci solitons
证明了如果两个梯度Ricci孤子沿各自的某个端点渐近于同一个正则锥，则孤子度量在这些端点的无穷远邻域上一定是等距的。我们的定理对度量的行为没有任何限制，特别是不需要它们的测地线完备性。作为应用，我们证明了R^n上唯一渐近于旋转对称锥的完全连通梯度收缩Ricci孤子是高斯孤子。
Backwards uniqueness for Mean curvature flow with asymptotically conical singularities
本文研究了具有渐近锥形奇点的均曲率流的逆唯一性，证明了在特定条件下两个流在奇点时间后完全一致，并展示了低熵流的逆唯一性应用。
Closed mean curvature flows with asymptotically conical singularities
本文利用Ważewski盒论证证明任意渐近锥形自收缩子对应存在形成对应奇点的闭超曲面平均曲率流，结合已有成果推出光滑闭超曲面水平集流肥化的存在性，为Evans–Spruck问题提供例证。
We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
Conical structure for shrinking Ricci solitons
证明了在无穷远处具有收敛于零的里奇曲率的收缩里奇孤子是渐近圆锥的。
On asymptotically conical curve shortening flows with convex projections
研究三维空间中具有凸投影的渐近锥形曲线在曲线缩短流下的长期演化行为。
Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities
研究具有锥奇点的紧致解析空间上Kähler-Ricci流的曲率界和渐近行为，证明流在奇点附近由唯一渐近于对应锥的Kähler-Ricci扩张孤子建模。
In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist comp…
We construct Gaussian Harmonic forms of finite Gaussian weighted $L^2$-norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers in $[11]$ under the existence of such ends. We show that the Morse index of a self-shrinker is greater or equal to $\frac{2g+r-1}{3}$, where $r$ is the number of asymptotically conical ends.
In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
Backward propagation of warped product structures and asymptotically conical shrinkers
本文研究Ricci流中扭曲积结构的向后传播，并证明渐近锥形梯度收缩孤子若渐近于Einstein流形乘积锥，则其自身为多重扭曲积。
We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying suff…
We study the degenerations of asymptotically conical Ricci-flat Kähler metrics as the Kähler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kähler metrics converge to a incomplete smooth Ricci-flat Kähler metric away from a compact subvariety. As a consequence, we construct singular Calabi-Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our resul…
Positive mass theorem for asymptotically flat spin manifolds with isolated conical singularities
本文证明了具有孤立锥形奇点或角状奇点的渐近平坦自旋流形的正质量定理，并允许非球面截面的拓扑奇点，从而涵盖负质量史瓦西度量情形。
Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces
考虑一类具有圆锥奇点、尖点和正则端点的二维非紧曲面的反问题。我们的激励例子是一个黎曼曲面$\mathcal M = Γ\backslash{\bf H}^2$与第一类富克斯群$Γ$包含抛物元素。那么$\mathcal M$是非紧的，并且有有限个尖点和椭圆奇点，它被认为是双曲轨道。引入了一类在有限部分上具有圆锥奇点，在无穷远处具有尖点和正则端点，度量是渐近双曲的黎曼曲面。通过观察Helmholtz方程在尖点处的解，定义了广义S-矩阵。然后，我们表明，这个广义S-矩阵决定的黎曼度量和结构的锥形奇点。
Desingularization of G_2 manifolds with isolated conical singularities
本文提出了一种对具有孤立锥奇点的紧致G_2流形进行去奇异化的方法，即通过切掉每个奇点的邻域，并将其粘在渐近锥G_2流形上。通过控制重叠粘合区域上的误差，我们可以利用Joyce的一个结果，得到紧致光滑7-流形具有无挠的G_2结构，并且具有完全的G_2完整性。由于渐近锥G_2流形的3次和4次上同调在每个锥奇点处都是粘在一起的，所以这一过程存在拓扑上的障碍。当具有孤立锥奇点的流形满足一定的必要拓扑条件时，我们可以在胶合过程中引入修正项，以保证胶合过程仍然有效。在4度障碍的情况下，这些校正项是平凡的构造，但在3度障碍的情况下，我们需要解决一个非紧流形上的椭圆型方程。为此，我们使用Lockhart-McOwen理论的加权索伯列夫空间流形的结束。在适当的规范固定条件下，利用该理论得到了渐近圆锥G_2流形上G_2结构的一个良好的渐近展开，这是使胶合程序工作所必需的.
Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow
研究渐近锥形梯度Kahler-Ricci扩张子上Kahler-Ricci流解的长时间行为与初始数据在空间无穷远处渐近行为的关系。
Positive mass theorem for asymptotically flat manifolds with isolated conical singularities
本文在不假设自旋条件的情况下，利用Schoen的共形爆破技术，证明了带有限多个孤立锥形奇点的渐近平坦流形的正质量定理。
We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.
Deformation Theory of Asymptotically Conical Coassociative 4-folds
我们研究了R ^[7]中的共结合4重N，它们对于速率λ <1的锥C是渐近圆锥的。若λ在区间[-2，1）中且为泛函数，证明了N的协结合变形的模空间是一个光滑流形，并计算了它的维数.当λ <-2且通有时，我们证明了模空间局部同胚于光滑流形间光滑映射的核，并给出了其期望维数的下界.我们还得到了一个测试时，N将是平面的，如果λ <-2，我们讨论的例子渐近圆锥coassociative 4倍。
Sp(2)-invariant expanders and shrinkers in Laplacian flow
本文研究 Laplacian 流中 Sp(2)-不变扩张子的分类与渐近锥性质，并猜想存在 SU(3)-不变扩张子，暗示流可实现拓扑手术。
A natural approach to the construction of nearly G2 manifolds lies in resolving nearly G2 spaces with isolated conical singularities by gluing in asymptotically conical G2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G2 manifolds, whose endpoint is the original nearly G2 conifold and whose parameter is the scale of the glued in asymptotically conical G2 manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical G2 manifolds: if the rate of t…
Ricci flow on asymptotically conical surfaces with nontrivial topology
作为有限全曲率的完备曲面上的里奇流的一般性研究的一部分，我们研究了具有渐近圆锥（包括作为特例的渐近欧氏）几何的曲面上的里奇流.在建立了长时间存在性，特别是流保持渐近圆锥几何的事实之后，我们证明了解度量$g(t)$以局部一致线性速率扩展;此外，重新缩放的度量族$t^{-1}g(t)$在无限时间处表现出转变，因为它局部一致地收敛到完整的，有限区域双曲度量是初始度量$g_0$的共形类中唯一的一致化度量。
Asymptotically Conical Associative 3-folds
给定R^7中的一个结合3重，它是渐近圆锥的且通有率小于1，我们证明了它的变形模空间局部同胚于光滑流形之间的光滑映射的核。此外，计算了模空间的虚维数，并表明对于大于-1的速率，虚维数是非负的，而对于小于或等于-1的速率，关联3倍预计是孤立的。
We give a complete description of the behaviour of Calabi-Yau instantons and monopoles with an $SU(2)^2$-symmetry, on Calabi-Yau 3-folds with asymptotically conical geometry and $SU(2)^2$ acting with co-homogeneity one. We consider gauge theory on the smoothing and the small resolution of the conifold, and on the canonical bundle of $\mathbb{CP}^1 \times \mathbb{CP}^1$, with their known asymptotically conical co-homogeneity one Calabi-Yau metrics, and find new one-parameter families of invariant instantons. We also entirely classify the relevant moduli-spaces of instantons and monopoles satisf…
We consider the moduli space $\mathcal{M}_ν$ of torsion-free, asymptotically conical (AC) Spin(7)-structures which are defined on the same manifold and asymptotic to the same Spin(7)-cone with decay rate $ν<0$. We show that $\mathcal{M}_ν$ is an orbifold if $ν$ is a generic rate in the non-$L^2$ regime $(-4,0)$. Infinitesimal deformations are given by topological data and solutions to a non-elliptic first-order PDE system on the compact link of the asymptotic cone. As an application, we show that the classical Bryant-Salamon metric on the bundle of positive spinors on $S^4$ has no continuous d…
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.
Isoperimetric structure of asymptotically conical manifolds
研究了渐近于具有非负里奇曲率的锥的黎曼流形的等周结构。具体地说，我们推广到这个设置的开创性成果G。Huisken和S.- T. Yau关于渐近平坦流形无穷远点上保体积稳定常中曲率曲面的正则叶状的存在性的结果以及第二作者与S. Brendle和J. Metzger关于渐近平坦流形的等周结构。我们还包括一个观察等周锥角等流形。这个结果是正质量定理在这种情况下的自然类比。
We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we produce, which will be always less or equal than one (with equality under suitable curvature or genericity assumptions), as well as of their precise asymptotic behaviour. In fact, we can prove that in any such surface for every couple of opposite half-lines there exists an embed…
An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for $t=0$
本文研究了渐近圆锥托里克收缩梯度凯勒-里奇孤立子的 Aubin 连续路径，证明了初始值参数的开放性及 t=0 的解。
We prove an expansion theorem on scalar-flat asymptotically conical Kähler metrics. Consider an AC Kähler manifold with asymptotic to a Ricci-flat Kähler metric cone with complex dimension n. Assuming the weak decay conditions required for the mass to be well-defined, then each scalar-flat AC Kähler metric admits an expansion that the main term is given by the standard Kähler metric of the metric cone and the leading error term is of O(r^{2-2n}) with coefficient only depending on the ADM mass and its dimension. Besides, the mass formula by Hein-LeBrun also can be proved in our setting. As an a…
Regularity of asymptotically conical Ricci-flat Kähler metrics
利用A. Grigor'yan和L. Saloff-Coste我们证明了在一个流形上的圆锥形结束的热核有一个高斯界。这一结果适用于渐近锥Kähler流形。这是作者和R.证明了Ricci-平坦Kähler锥的一个可分解在每一个Kähler类中都存在一个Ricci-平坦Kähler度量渐近于锥度量。我们证明了该度量到锥度量收敛速率。对于紧致Kähler类，这与P. Kronheimer和D. Joyce.
We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.
On conical asymptotically flat manifolds
本文证明了Petrunin和Tuschmann关于不存在渐近于半平面的渐近平坦4-流形的猜想，并综述了作为研究动机的引力瞬子相关进展与问题。
Asymptotically conical Calabi-Yau manifolds, III
在最近的预印本中，池莉证明了无穷远处具有正则切锥的渐近锥复流形允许全纯紧化（他的结果很容易推广到拟正则情形）。本文证明了如果开流形是Calabi-Yau流形，则Chi Li紧化是投射代数的。这有两个应用。首先，每一个这类的Calabi-Yau流形都可以用我们在本系列第二篇文章中的精化的Tian-Yau型定理来构造。其次，我们证明了分类定理，这种流形通过变形的法锥。这包括Kronheimer对ALE空间的分类和Stenzel度量的唯一性定理。
Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities
该论文作为平均曲率流一般动力学系列的第二篇，针对首次奇点为渐近锥形收缩子的平均曲率流初始扰动展开研究，引入费曼-卡茨公式和非紧不变锥方法，证明一般初始扰动可使重标流避免锥形奇点。
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