David Pumberger, Tristan Riviere
We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex manifolds and deduce the uniqueness of their tangent maps.
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Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces
该研究针对具有适当小奇点集的浸入稳定极小超曲面，建立了带孤立奇点的切锥的唯一性与正则性结果，且该结果允许切锥具有任意整数重数。
We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and an almost-monotonicity formula for the mass at boundary points.
We consider $2$-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents in the euclidean space. This note is also the first step in a regularity program for semicalibrated $2$-dimensional currents and spherical cross sections of $3$-dimensional area minimizing cones.
Uniqueness of Cylindrical Tangent Cones $C_{p,q} \times \mathbb{R}$
研究面积最小超曲面在 $\mathbb{R}^9$ 中柱形切锥 $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ 的唯一性，完善了 Simon 和 Székelyhidi 对高维切锥唯一性的证明。
We proved a uniqueness theorem of tangent connections for a Yang-Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained controls over the rate of the asymptotic convergence of the connection to the tangent connection under assumptions that the connection is stationary or the tangent connection is integrable. There are parallel results for the cones at infinity of a Yang-Mills connection on an asymptotically flat manifold. We also gave an application of our methods to the Yang-Mills flow and proved that the Yang-Mills flow exi…
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…
We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section.
This part II of the paper is concerned with questions of existence and uniqueness of tangents in the special case of G-plurisubharmonic functions, where G is a compact subset of the Grassmannian of p-planes in ${\mathbb R}^n$. An upper semi-continuous function u on an open set $Ω$ in ${\mathbb R}^n$ is G-plurisubharmonic if its restriction to $Ω\cap W$ is subharmonic for every affine G-plane $W$. Here G is assumed to be invariant under a subgroup K of O(n) which acts transitively on the sphere $S^{n-1}$. Tangents to u at a point x are the cluster points of u under a natural flow (or blow-up) a…
There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the "Riesz characteristic", which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-…
Uniqueness in the Plateau problem for calibrated currents
该研究针对校准流的有向Plateau问题，结合面积最小化流的边界正则性理论与Frank Morgan风格的唯一延拓论证，证明了具连通$C^{3,α}$边界的紧支撑光滑校准积分流的解唯一性，还推广至连续校准积分平链，并在余维1情形给出以校准形式正则性表述的唯一性充分条件。
Let $(M, \om)$ be a symplectic manifold, endowed with a compatible almost complex structure J and the associated metric g . For any p \in {1, 2, ... (dim M)/2} the form $\Om := \frac{\om^p}{p!}$ is a calibration. More generally, dropping the closedness assumption on $\om$, we get an almost hermitian manifold $(M, \om, J, g)$ and then $\Om$ is a so-called semi-calibration. We prove that integral cycles of dimension 2p (semi-)calibrated by $\Om$ possess at every point a unique tangent cone. The argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem…
We partially resolve a conjecture of Meeks on the asymptotic behavior of minimal surfaces in $\mathbb{R}^3$ with quadratic area growth.
We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings. Furthermore, given such a closed, multiplicity-one, smoothly embedded self-similarly shrinker $Σ$, we show that any solution of the rescaled flow, which is sufficiently close to $Σ$, with Gaussian density ratios greater or equal to that of $Σ$, stays for…
In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzi to observe a uniqueness property for tangent flows at the first singular time of a smooth mean curvature flow of a closed surface in 3-dimensional Euclidean space. Specifically, if, at a fixed singular point, one tangent flow is a positive integer multiple of a shrinking plane, cylinder or sphere, then, modulo rotations, all tangent flows at the point are the same.
On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity
本文证明任意整数边界重数的面积最小化流在密度 $Q/2$ 边界点处切锥唯一且收敛具有幂次速率，并由此导出支撑高重数边界正则性理论的分解定理。
Uniqueness of tangent flows in free boundary flow
该研究证明了自由边界平均曲率流在半圆柱自收缩体奇点处的切流唯一性，补充了圆柱奇点唯一性理论的自由边界类比。
We show that the cylindrical tangent cone $C\times \mathbf{R}$ for an area-minimizing hypersurface is unique, where $C$ is the Simons cone $C_S= C(S^3\times S^3)$. Previously Simon proved a uniqueness result for cylindrical tangent cones that applies to a large class of cones $C$, however not to the Simons cone. The main new difficulty is that the cylindrical cone $C_S\times \mathbf{R}$ is not integrable, and we need to develop a suitable replacement for Simon's infinite dimensional Lojasiewicz inequality in the setting of tangent cones with non-isolated singularities.
Almost minimizing Yang$-$Mills fields: log-epiperimetric inequality, non-concentration, and uniqueness of tangents
本文为任意维度Yang-Mills场建立直接的对数周长不等式，并据此证明能量极小化Yang-Mills场和$ω$-ASD连接的孤立奇点切锥唯一性，同时通过Luckhaus型引理排除blow-up序列的曲率集中。
The Fine Structure of the Singular Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points and $\mathcal{H}^{m-2}$-a.e. Uniqueness of Tangent Cones
本文证明了光滑 Riemannian 流形中余维数大于1的面积最小化整流单体在 $\mathcal{H}^{m-2}$-几乎处处具有唯一的切锥，并结合先前工作得出其奇异集是可数 $(m-2)$-可求长的。
Uniqueness of tangent cones with immersed special Legendrian links
该研究在特定假设下证明了$\boldsymbol{\text{C}}^3$中带有横截二重极点的紧浸入特殊拉格朗日链的特殊拉格朗日切锥的唯一性，核心方法基于离散Lojasiewicz型不等式，源于胶合障碍机制。
We show that if an exact special Lagrangian $N\subset \mathbb{C}^n$ has a multiplicity one, cylindrical tangent cone of the form $\mathbb{R}^{k}\times \mathbf{C}$ where $\mathbf{C}$ is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $\mathbf{C}$ satisfies an integrability condition. This applies, for example, when $\mathbf{C}= \mathbf{C}_{HL}^{m}$ is the Harvey-Lawson $T^{m-1}$ cone for $m\ne 8,9$.
`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing technique. We have existence theorems in that sense, but there seems to have been no uniqueness theory for higher-dimensional ones such as special Lagrangian submanifolds, which we discuss in the present paper.
Let $(X, L)$ be a polarized Calabi Yau variety (or canonical polarized variety) with crepant singularity. Suppose $ω_{KE} \in c_1(L)$ (or $ω_{KE} \in c_1(K_X)$) is the unique Ricci flat current (or Kahler Einstein current with negative scalar curvature) with local bounded potential constructed in [18], we show that the local tangent at any point $p \in X$ of metric $ω_{KE}$ is unique
Given a smooth, symmetric, homogeneous of degree one function $f=f\left(λ_{1},\cdots,\,λ_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\, n$, and an oriented, properly embedded smooth cone $\mathcal{C}^n$ in $\mathbb{R}^{n+1}$, we show that under some suitable conditions on $f$ and the covariant derivatives of the second fundamental form of $\mathcal{C}$, there is at most one $f$ self-shrinker (i.e. an oriented hypersurface $Σ^n$ in $\mathbb{R}^{n+1}$ for which $f\left(κ_{1},\cdots,\,κ_{n}\right)+\frac{1}{2}X\cdot N=0$ holds, where $X$ is the position vector, $N$ is the unit nor…
We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be produced already in C^n, hence our result is optimal. The key idea is an implementation, for currents in an almost complex setting, of the classical blow up of curves in algebraic or symplectic geometry. Unlike the classical approach in C^n, we cannot rely on plurisubharmonic potenti…
The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varieties
本文证明了从任意近复流形到射影代数簇的弱全纯映射的切映射唯一性，并由此得到非可求长正伪全纯环的唯一切锥性质，同时为Bellettini关于正积分环的结果提供了新证明。
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.
Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow
本文在基点附近几何接近标准柱面或其商结构的假设下，建立了一个指向$\mathcal{W}$-熵的Lojasiewicz不等式，并证明了Ricci流首次奇异时间处柱状切流的强唯一性。
Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators
本文研究了均值曲率流翻译子在无限远处可能存在的非可移除奇点，并证明了有限熵、有限亏格、嵌入的坍缩翻译子在无限远处收敛到唯一确定的平面集合。
Uniqueness of tangent flows at infinity for finite-entropy shortening curves
本文证明了有限熵古代曲线缩短流在无穷远处切流的唯一性，并通过重标度流的指数收敛及对尖端、顶点、拐点和图形半径增长率的精确刻画，排除了收缩圆、静态直线、回形针和平移冷酷收割者等例外情形。
We consider the Calabi-Yau metrics on $\mathbf{C}^n$ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone $\mathbf{C}\times A_1$ at infinity for the $(n-1)$-dimensional Stenzel cone $A_1$. We show that up to scaling and isometry this Calabi-Yau metric on $\mathbf{C}^n$ is unique. We also discuss possible generalizations to other manifolds and tangent cones.
Cylindrical tangent flows in mean curvature flow
本文针对平均曲率流的柱形奇点，采用受Székelyhidi启发的新方法，证明其唯一性与刚性结果，完善了Colding-Minicozzi相关研究的结论
Uniqueness of blowups for forced mean curvature flow
本文证明了受迫平均曲率流在闭自收缩子和圆圆柱自收缩子处切锥的唯一性，推广了Schulze和Colding-Minicozzi的结果，并涵盖黎曼流形情形。
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $μ$-polystable holomorphic bundles over $\mathbf{P}^{n-1}$.
Singularity formations in Lagrangian mean curvature flow
该文研究拉格朗日平均曲率流中由重数1的光滑特殊拉格朗日锥给出切流的奇点，得到多类唯一性与爆破极限存在性结果，还推广了Lawlor neck的唯一性定理，并应用于横截平面切流奇点的建模。
Non-uniqueness of Brakke flows starting from minimal surfaces with singularities
该研究证明了满足特定条件的带奇点极小曲面出发的Brakke流存在非唯一性，且无需假设奇点处切锥的唯一性，得到了相关曲面的动力学不稳定性。
We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian-Yang-Mills connections.
It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension 4 with non-Euclidean volume growth who has at least two distinct tangent cones at infinity and one of them has a smooth cross section.
Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow
本文在广义圆柱体假设下扩展了Ricci流中指态W熵的Lojasiewicz不等式，证明了广义圆柱切流的强唯一性，并展示了由某些切流给出R^k×N^{n−k}或其商的子集S^k_qc(N)是水平抛物线k-可测的。
Analysis of singularities of area-minimizing currents, Part II: a uniform height bound, estimates away from branch points of rapid decay, and uniqueness of tangent cones
本文为n维面积最小化可求长流T建立新框架，给出高度估计和衰减估计。结合第一部分定理，得出T在几乎每个\({\mathcal H}^{n - 2}\)点有唯一切锥，其奇点集可数\((n - 2)\)可求长，分支点一般快速衰减到唯一切平面。
Symmetric log-epiperimetric inequality for harmonic maps with analytic target and applications
本文建立解析目标流形上调和映射的对称（对数）超周边不等式，并据此重新证明孤立奇点切映射的唯一性，同时得到适当能量增长条件下无穷远切映射的唯一性。
We investigate the maximal open domain $\mathscr{E}(M)$ on which the orthogonal projection map $p$ onto a subset $M\subseteq \mathbb{R}^d$ can be defined and study essential properties of $p$. We prove that if $M$ is a $C^1$ submanifold of $\mathbb{R}^d$ satisfying a Lipschitz condition on the tangent spaces, then $\mathscr{E}(M)$ can be described by a lower semi-continuous frontier function. We show that this frontier function is continuous if $M$ is $C^2$ or if the topological skeleton of $M^c$ is closed and we provide an example showing that the frontier function need not be continuous in g…
We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dimensional planes, and in a neighbourhood of any point at which at which at least one tangent cone is equal to a union of four distinct multiplicity one n-dimensional half-planes that meet along an (n-1) - dimensional axis. The key ingredient is a new Excess Improvement Lemma obtai…
A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energies
本文通过提供高斯映射路径长度的精确下界，证明了关于高斯映射的芬谢尔定理，并展示圆圈在大多数广义切点能量下是最小化的，同时证明盘在所有分数Willmore能量下是凸平面集中的最小化解。
In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by ReΩin a Calabi-Yau n-fold (M,Ω), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X,T) is a stationary Lagrangian integer multiplicity current in R\sup 2n with volume density greater than one at X. When n=2, the tangent cone consists of a finite union of more than one 2-planes in R\sup 4 which are complex in a complex structure on R\su…
Corrections to a paper of Allard and Almgren on the uniqueness of tangent cones
针对Allard和Almgren1981年切锥唯一性论文中的不等式错误及定义、证明缺失等问题，研究者重写了原论文5.1-5.4节，补充了相关必要内容，给出了技术细节完善的修正版本。
We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.
We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called $k$-dimensional $(\varepsilon,R)$ Reifenberg flat sets in $\mathbb{R}^n$. Our results generalize the ones from a previous paper by the author, in which the co-dimension one case (i.e. $k=n-1$) was studied. For $\varepsilon$ fixed, this class is general enough to include (i) all $C^2$ sub-manifolds (ii) all Lipschitz sub-manifolds with Lipschitz constant less than $\varepsilon$ (iii) some sets with Hausdorff dimension larger than $k$. The Reifenberg condition, r…
Geometry of Bounded Frechet Manifolds
本文发展了有界Fréchet流形的几何，证明其二阶切丛成为向量丛当且仅当被赋予线性联络，并据此证明了向量场积分曲线的存在性与唯一性。
Donaldson-Sun Theory in the Conic Case
将Donaldson-Sun度量切锥理论推广到边界系数为有理数有限子集的锥Kähler-Einstein对的非塌缩Gromov-Hausdorff极限，证明对数度量切锥的唯一性，关联稳定退化机制，构造反例表明Kähler-Einstein假设对切锥刚性的必要性。
On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $Ω$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $Ω$-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when $n=1$ and $Ω$ is the volume element of $\bar{M}$. To the second variation we associate an $Ω$-Jacobi operator and define $Ω$-stablility. Under natural conditions, we prove that the Euclidean $m$-spheres are the unique $Ω$-s…
This is the first part in a two-part series on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition $O(r^{-n-ε})$ needed in earlier work to $O(r^{-ε})$, relying on some new ideas about harmonic functions. We then look at a few examples: (1) Crepant resolutions of cones. This includes a new class of Ricci-flat small resolutions associated with flag manifolds. (2) Affine deformations of cones. One focus here is the question of the precise rate of decay of the metric …
We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov-Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop-Gromov volume comparison theorem and Perelman's celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas a…
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