Brian White
Shoen and Uhlenbeck showed that ``tangent maps'' can be defined at singular points of energy minimizing maps. Unfortunately these are not unique, even for generic boundary conditions. Examples are discussed which have isolated singularities with a continuum of distinct tangent maps.
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We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, or, more generally a directed curve or a frontal curve which has well-defined tangent directions along the curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the c…
It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and differential systems on flag manifolds, via several techniques in differentiable algebra. It is provided also the generic diffeomorphism classification of singularities on tangent varieties to contact-integral curves in the standard contact projective space. Moreover we give basic results on the classification of singularities of tangent varieties to generic …
It is known that the class of developable surfaces which have zero Gaussian curvature in three dimensional Euclidean space is preserved by the parallel transformations. A tangent developable surface is defined as a ruled developable surface by tangent lines to a space curve and it has singularities at least along the space curve, called the directrix or the the edge of regression. Also the class of tangent developable surfaces are invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalized for frontal curves…
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 $\mathbf{P}^{n-1}$.
We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umeh…
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphism classification is completed for generic directed curves. Then it turns out that the swallowtails and open swallowtails appear generically for the classification on singularities of tangent surfaces.
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
This is a survey article on recognition problem of frontal singularities. We specify geometrically several frontal singularities and then we solve the recognition problem of such singularities, giving explicit normal forms. We combine the recognition results by K. Saji and several arguments on openings, which was performed for the classification of singularities of tangent surfaces (tangent developables) by the author. As an application of our solutions of recognition problem of frontal singularities, we announce the classification of singularities appearing in tangent surfaces of generic null…
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。
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 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。
On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces
本文研究 Hirzebruch 曲面上塌缩的 Kähler-Ricci 流的有限时间奇点，证明其切流对应于具有有限多个 orbifold 奇点的非平坦梯度 Kähler-Ricci 收缩孤立子。
This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of …
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…
In this paper, Legendre curves on unit tangent bundle are given using rotation minimizing (RM) vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classifed.
Singularities of Mean Curvature Flow with Bounded Total Curvature
该研究在总曲率有界的条件下，证明了光滑闭嵌入平均曲率流首次奇异时刻的切流具有光滑支撑，将Ilmanen的相关结果推广到更高维。
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.
We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold $S$ changes its signature (degenerates) along a curve $S_0$, which locally separates $S$ into a Riemannian ($R$) and a Lorentzian ($L$) domain. The geodesic flow does not have singularities over $R$ and $L$, and for any point $q \in R \cup L$ and every tangential direction $p$ there exists a unique geodesic passing through the point $q$ with the direction $p$. On the contrary, geodesics canno…
In a previous work, the authors introduced the notion of `coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss-Bonnet formulas on coherent tangent bundles on 22-dimensional manifolds were proven, and several applications to surface theory were given. Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal E$ be a vector bundle of rank $n$ over $M^n$, and $φ:TM^n\to \mathcal E$ an oriented vector bundle homomorphism. In this paper, we sh…
We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity of envelopes (two transversal semicubic cusps), and we describe its perestroikas under generic small deformations.
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。
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。
In Part I of this paper we have seen that any singular compact area minimizer in a positive scalar curvature manifold admits a conformal deformation to some minimal factor geometry that shares many properties with the minimizer, like the Ahlfors regularity, the validity of Poincare inequalities and the presence of tangent cones with positive scalar curvature. In this Part II we show that these geometries admit surgery style arguments eliminating the singular sets.
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…
Rigidity properties of holomorphic Legendrian singularities
本文在复解析范畴中研究接触流形的勒让德奇点，证明了约化切锥奇点与切锥的接触同胚双全纯等价性，以及正规勒让德奇点的形变刚性，方法基于无穷小接触同胚与自然线丛截面的联系。
In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front, which is a generalization of a theorem of O'Neill and Stiel: any isometric immersion of the n-sphere into the (n+1)-sphere of the same sectional curvature is totally geodesic.
We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels. On an appropriate non- singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel. When the rough metric induces a metric measu…
Which singular tangent bundles are isomorphic?
研究对数切丛和b-切丛与标准切丛或其他奇异切丛同构的条件，分析低维情形和球面情形，并建立b^m-切丛的Poincaré-Hopf定理。
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。
Deformations of harmonic maps with conical singularities
本文研究紧致流形间具有锥形奇点的调和映射的形变理论，在切映射满足特定指标和非退化条件下证明其在度量扰动下保持，并构造了满足条件的显式例子。
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 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。
In this note we introduce the notion of a smooth structure on a conical pseudomanifold $M$ in terms of $C^\infty$-rings of smooth functions on $M$. For a finitely generated smooth structure $C^\infty (M)$ we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of $M$, and the notion of characteristic classes of $M$. We prove the vanishing of a Nash vector field at a singular point for a special class of Euclidean smooth structures on $M$. We introduce the notion of a conical symplectic form on $M$ and show that it is smooth with respect to a Euclidean…
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 construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work of Hein-Naber.
Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$
本研究针对任意余维数下的$m$维模$(q)$面积极小流，通过精细过剩衰减定理与通用频率函数几乎单调性，刻画了其内部奇异集的两类精细结构，推进了几何测度论中模极小流奇点正则性的研究。
Nonproduct Minimal Submanifolds with Cylindrical Tangent Cones
本文通过扰动方法构造无穷维族具有柱形切锥奇异集的非积极小子流形，解答了Leon Simon的问题，推广了Caffarelli-Hardt-Simon的工作，需用tame Fréchet空间范畴与Nash-Moser隐函数定理完成研究。
Generic Metrics on $S^{n+1}$ Preclude Linearly Stable Singular Tangent Cones of Area-Minimizing Boundaries
论文证明在 $S^{n+1}$ 上的 $C^{3} $ 拓扑下，残余集中的度量下，无面积最小积分 $n$-流体具有线性稳定的奇异性切锥。通过扰动定理、开放集性质和巴尔-类别论证，得出结论。
On Cylindrical singularities of minimal hypersurfaces
本文在$\text{R}^9$中构造了一类极小超曲面，其奇异集含原点及收敛于原点的孤立奇点，关键步骤是构造了具有特定切锥性质的整体稳定超曲面。
On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds
本文研究了具有积分平均曲率界的弱平均曲率流的奇点结构，证明了任意切向流由平稳锥给出，并在特定条件下证明了切向流的唯一性，同时揭示了在面积最小化 Simons 锥点处存在光滑 Hardt-Simon 极小曲面伴随极限流。
Any ruled surface in Euclidean 3-space is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra of type (0,3,1)). Combining this classical framework and Singularity Theory, we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invariants. In particular, using a theorem of G. Ishikawa, we show that local topological type of singular (non-cylindrical) developable surfaces is completely determined by vanishing order of the dual torsion, that generalizes an old result of D. Mond for tangent developables of non-…
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.
A 2-web in the plane is given by two everywhere transverse 1-foliations. In this paper we introduce the study of singular 2-webs, given by any two foliations, which may be tangent in some points. We show that such two foliations are tangent along a curve, which will be called the polar curve of the 2-web, and we study the relationship between the contact order of leaves of both foliations and the singularities of the polar curve.
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 …
Special Lagrangians with multiple isolated singularities
该研究将Caffarelli-Hardt-Simon扰动论证推广到特殊拉格朗日量领域，证明正则特殊拉格朗日锥的桥接原理，进而得到具有多个给定孤立锥形奇点的特殊拉格朗日子流形的存在性。
Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program
该研究针对解析极小模型纲领中的凯勒-里奇流有限时间奇点，在特定假设下证明复二维非坍缩凯勒-里奇流奇点行为的模型，在高维及紧致收缩子情形也获相关结果，完善了相关理论。
Isolated singularities of harmonic maps with generic boundary data
本文研究具有一般边界数据的调和映射孤立奇点，提出最小指标原理，证明在低维情况下奇点切映射为径向投影，并建立无穷远刚性定理。
Asymptotic Analysis of Harmonic Maps With Prescribed Singularities
本文作为系列论文首篇，研究带规定奇点的调和映射的渐近行为，为建立多黑洞质量-角动量不等式提供正则性理论，还用于证明极端黑洞近视界极限存在并计算其几何。
For a smooth surface in $\mathbb{R}^3$ this article contains local study of certain affine equidistants, that is loci of points at a fixed ratio between points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the equidistant contains one or other point of contact). The situation studied occurs generically in a 1-parameter family, where two parabolic points of the surface have parallel tangent planes at which the unique asymptotic directions are also parallel. The singularities are classified by regarding the equidistants as critical values of a 2-parameter unfolding of…
We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, where P' is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plan…
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…
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。
Ricci flow from spaces with edge type conical singularities
本文研究沿闭嵌入曲线具有边型锥奇点的空间上的Ricci流，在局部模型为正曲率锥乘直线的假设下，证明了解的存在性、收敛性、曲率估计及切流为膨胀孤子乘直线。
Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities
本文针对具有孤立奇点的常平均曲率超曲面，证明了在切锥正则且超曲面为最小化Caccioppoli集边界时，存在一列同平均曲率的光滑超曲面收敛到它，并指出维数8时锥条件可省略。
When the rank of the bundle is $\geq 2$, in a certain sense, we found an essential obstruction for the gluing construction of $G_{2}-$instantons with $1-$dimensional singularities. It involves the Atiyah classes generated by contracting a vector in $\mathbb{C}^{3}$ with the curvature. Intuitively speaking, the gluing does not work if the tangent connection at a component of the $1-$dimensional singular locus is not the twisted Fubini-Study connection on a twisted tangent bundle of $\mathbb{P}^{2}$. Particularly, it fails if the rank of the bundle is $\geq 3$.
We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ parallel to the tangent field of $f_0$, find a conformally parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve as a singular set and such that the partial derivatives $f_x$ and $f_y$ are given by $\frac{\dd f_0}{\dd x}$ and $v$ along the curve. Within the clas…
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