Joel B. Mohler, Ron Umble
Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geodesic. Minimal paths are geodesics in the sense of Banchoff.
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Singularities of minimal submanifolds
本文综述极小曲面奇点结构的关键结果与未决问题，并讨论稳定极小超曲面奇点研究中的技术简化。
Singularities of generalized timelike minimal surfaces in Lorentz-Minkowski 3-space
本文研究Minkowski 3维空间中类时极小曲面的奇点，证明其存在性与不存在性定理，给出多类奇点的判别准则、对偶与不变性定理及例子。
In the 3-dimensional Lorentz-Minkowski space we prove that the sign of the Gaussian curvature of any timelike minimal surface is determined by the degeneracy and the orientations of the two null curves that generate the surface. Moreover, we also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface which admits some kind of singular points.
Rigidity of Minimal Surfaces with Y-singularities and low Morse Index in $\mathbb{R}^3$
本文研究带 Y 奇点且低 Morse 指数的极小曲面，通过刚性结果给出指数为一情形的部分唯一性分类，突出 Y-catenoid 的特殊地位。
Singularities on timelike minimal surfaces in Lorentzian Heisenberg group
研究了洛伦兹希格纳群中时状极小曲面的构造及其奇点特性，通过调和映射和具体例子探讨了不同奇点类型。
Singularities of Discrete Indefinite Affine Minimal Surfaces
本文研究离散不定仿射极小曲面的奇点，通过离散Lelieuvre公式定义离散渐近网的奇异边和顶点，使光滑情形中奇异集的关键性质在离散情形下仍然保持。
This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical boundary value problems and we also consider eigenvalue problems and variational problems on such singular spaces.
A class of affine maximal surfaces with singularities and its relationship with minimal surface theory
本文定义了带奇点的仿射极大曲面子类 affine maxfaces，证明其满足 Osserman 型不等式，并揭示了其与 Euclidean 极小曲面理论的关系。
In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also includes statements about the structure of compact manifolds of positive scalar curvature extending the work of \cite{sy1} to all dimensions. The technical work in this paper is to construct minimal slicings and associated weight functions in the presence of small singular sets and to show that the singular sets do not become too large in the lower dimensional…
Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and to characterize them using a geometric invariant called lightlike curvatures. As a result, we reveal the relationship between timelike Thomsen surfaces, and timelike minimal surfaces with planar curvature lines. As an application, we give a deformation of null curves preserving …
We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces $(Σ,g_1)$ and $(Σ,g_2)$ when the cone angles of $g_1$ and $g_2$ are different and smaller than $π$. When the cone angles of $g_1$ are strictly smaller than the ones of $g_2$, this minimal diffeomorphism is unique.
We show that to every maximal surface with conelike singularities in Lorentz-Minkowski space $\mathbb{L}^3$ that can be locally represented as the graph of a smooth function, there exists a corresponding timelike minimal surface in $\mathbb{L}^3$. There exists a linear transformation between such a maximal surface and its corresponding timelike minimal surface and it maps the singularities of one to the singularities of the other. Moreover, this transformation establishes a one-one correspondence between such maximal surfaces and timelike minimal surfaces and also preserves the one-one propert…
Given a compact closed subset $M$ of a line segment in $\mathbb{R}^3$, we construct a sequence of minimal surfaces $Σ_k$ embedded in a neighborhood $C$ of the line segment that converge smoothly to a limit lamination of $C$ away from $M$. Moreover, the curvature of this sequence blows up precisely on $M$, and the limit lamination has non-removable singularities precisely on the boundary of $M$.
We prove that there does not exist global-in-time axisymmetric solutions to the time-like minimal submanifold system in Minkowski space. We further analyze the limiting geometry as the maximal time of existence is approached.
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 …
Removable singularities for nonlocal minimal graphs
本文研究了非局部极小图在特定条件下的可移除奇点问题，证明了在满足容量为零的条件下，非局部极小图可扩展至整个区域。
By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.
Removing singularities of minimal surfaces by isotopies
针对开黎曼曲面到高维欧氏空间的共形极小曲面，证明其分支点与有限全曲率完备端可通过同痕移除，该结论对到复空间的零全纯曲线同样成立。
On boundedness of singularities and minimal log discrepancies of Kollár components, II
本文证明K-半稳定log Fano锥奇点的有界性由局部体积下界和Kollár分量极小对数差异上界决定，并据此确认了三维情形的有界性猜想及klt奇点局部体积的聚集性质。
Singly periodic maximal graphs with isolated singularities in Lorentz-Minkowski 3-space
利用Weierstrass表示法构造了具有孤立锥状奇点的单周期极大曲面，探讨了其在洛伦兹-闵可夫斯基3空间中的性质。
Minimal submanifolds with multiple isolated singularities
本文将Smale的奇异桥原理推广至任意余维数及$n \geq 3$，并利用Lawson-Osserman锥在$\mathbb{R}^7$中构造出具有任意有限数目孤立奇异点的四维极小图。
In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete surfaces with non-zero constant Gaussian curvature, and parallel surfaces of discrete minimal and maximal surfaces, and discrete constant mean curvature $1$ surfaces in de Sitter $3$-space, including comparisons with different previously known definitions of such singularities.
The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Koll'ar's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.
Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to refine the classification of maximal surfaces with planar curvature lines, but also to show that there exists a deformation consisting exactly of all such surfaces. Furthermore, we investigate the types of singularities that occur on maximal surfaces with planar curvature lines. Fi…
On boundedness of singularities and minimal log discrepancies of Kollár components
本文研究klt奇点的有界性问题，证明在三维或Kollár分量极小对数差异有上界时，局部体积远离零的klt奇点有界至多相差特殊退化，并回答了关于对数典范阈值与局部体积关系的一个问题。
We construct a sequence of compact embedded minimal disks in the unit ball in Euclidean 3-space whose boundaries are in the boundary of the ball and where the curvatures blow up at every point of a line segment of the vertical axis, extending from the origin. We further study the transversal structure of the minimal limit lamination and find removable singularities along the line segment and a non-removable singularity at the origin. This extends a result of Colding and Minicozzi where they constructed a sequence with curvatures blowing up only at the center of the ball, Dean's construction of…
Metrics with minimal singularities and the Abundance conjecture
本文研究了Abundance猜想，给出了当$χ(X,\mathcal O_X)\neq0$时基于极小奇异性电流多重理想渐近行为的充要条件，并证明了超规范电流的结构与正则性，指出其对完成该猜想证明的核心作用。
We discuss recent results on minimal surfaces and mean curvature flow, focusing on the classification and structure of embedded minimal surfaces and the stable singularities of mean curvature flow. This article is dedicated to Rick Schoen.
Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries
本文研究平均曲率流边界奇点处收缩子的可定向性，并构造出出现不可定向光滑嵌入收缩子的非空初始曲面开集；同时利用平均曲率流改进极小曲面边界总曲率的临界阈值。
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…
We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons w…
We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized su…
In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a Kähler-Einstein surface. We prove that the blow-up flow $Σ_s^\infty$ at a singular point $(X_0, T_0)$ of a symplectic mean curvature flow $Σ_t$ or of a Lagrangian mean curvature flow $Σ_t$ is a non trivial minimal surface in ${\bf R}^4$, if $Σ_{-\infty}^\infty$ is connected.
Observing minimal submanifolds with isolated singularities on open sets of metrics
针对度量开集上带孤立奇点的极小子流形，发展扰动理论并得到“零散持续性”准则，补充非持续性定理，构造出可在度量开集上观测的带孤立奇点的高维稳定 varifold 族。
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 symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.
In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divisorial contractions for varieties of general type. In case one has a smooth minimal model of general type (i.e., the canonical bundle is nef and big), we show infinite time existence and analyze the singularities.
Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup…
We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical {\it horizontal} area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces in the horizontal spherical bundle over the base manifold. Generic singularitie…
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。
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。
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