M. Haskins, N. Kapouleas
For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.
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We study special Lagrangian cones in $\C^n$ with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in $\C^3$ each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in $\C^3$ with a spherical link -- any such cone must be a plane. We also construct a one-parameter family of asymptotically conical special Lagrangian submanifolds from any special Lagrangian cone.
We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail the geometry of these $\sorth{p}\times \sorth{q}$-invariant SL cones, in preparation for their application to our higher dimensional special Legendrian gluing constructions. In particular the symmetries of these cones and their asymptotics near the spherical limit are analysed. All $\s…
Special Lagrangian cones in Deep Learning
本文提出Harvey-Lawson锥的矩阵推广，证明其为精确特殊拉格朗日流形，并属于叶状结构划分深度学习平衡流形的流形族。
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。
Self-expanders of positive genus
本文针对三维空间中的一般锥体构造正亏格自展开子，并用于构造平均曲率流，使亏格在奇异时刻不降为零，同时构造无界亏格序列并刻画其渐近行为。
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 $m\ne 8,9$.
We construct some examples of special Lagrangian submanifolds and Lagrangian self-similar solutions in almost Calabi-Yau cones over toric Sasaki manifolds. For example, for any integer g>0, we can construct a real 6 dimensional Calabi-Yau cone M_g and a 3 dimensional special Lagrangian submanifold L^1_g in M_g which is diffeomorphic to the product of a closed surface of genus g and the real line R, and a 3 dimensional compact Lagrangian self-shrinker L^2_g in M_g which is diffeomorphic to the product of the closed surface of genus g and a circle S^1.
We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this result by showing that connected components of the rescaled flow converge to an area-minimizing cone, as opposed to possible non-area minimizing union of Slag cones. In the last section, we give specific examples for which such singularity formation occurs.
As was shown by a part of the authors, for a given $(2, 3, 5)$-distribution $D$ on a $5$-dimensional manifold $Y$, there is, locally, a Lagrangian cone structure $C$ on another $5$-dimensional manifold $X$ which consists of abnormal or singular paths of $(Y, D)$. We give a characterization of the class of Lagrangian cone structures corresponding to $(2, 3, 5)$-distributions. Thus we complete the duality between $(2, 3, 5)$-distributions and Lagrangian cone structures via pseudo-product structures of type $G_2$. A local example of non-flat perturbations of the global model of flat Lagrangian co…
The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend the results on special Lagrangian submanifolds on $\mathbb C^n$ to the toric Calabi-Yau cones over Sasaki-Einstein manifolds.
Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture
本文构造了新的均曲率流扩张器，这些扩张器光滑地趋于某些收缩器产生的锥体。对于每个锥体，证明了存在任意高亏格的扩张器，反驳了伊尔曼恩的亏格减少猜想。
In this article I show that every special Lagrangian cone in C^3 determines, and is determined by, a primitive harmonic surface in the 6-symmetric space SU_3/SO_2. For cones over tori, this allows us to use the classification theory of harmonic tori to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some examples found recently by Joyce are put into this context.
We prove a number of results relating various measures (volume, Legendrian index, stability index, and spectral curve genus) of the geometric complexity of special Lagrangian $T^2$-cones. We explain how these results fit into a program to understand the "most common" three-dimensional isolated singularities of special Lagrangian submanifolds in almost Calabi-Yau manifolds.
We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth oriented Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a wea…
Special Lagrangians with Cylindrical Tangent Cones
研究构造了具有孤立奇点的特殊Lagrangian子流形，其切锥为圆柱形，且子流形连通而切锥不连通。
In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar ends like the Lagrangian catenoid. Then we show the existence of embedded minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we construct genus $g$ examples in $\mathbb{R}^4$ with $d$ embedded planar ends such that $g\geq 1$ and $g+2\leq d\leq 2g+1$. These examples include a family of embed…
Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces
本文证明具有锥奇点的闭球面之间的极小拉格朗日微分同胚必为等距映射，并将Liebmann刚性定理推广到常正高斯曲率闭曲面的分支浸入情形。
Existence and non-uniqueness of cone spherical metrics with prescribed singularities on a compact Riemann surface with positive genus
本文利用线丛的多稳定扩张，证明了正亏格紧致黎曼面上具预定奇点的锥球面度量在多种条件下的存在性与非唯一性。
Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one
本文研究非紧纤维完备 Lagrange 纤维化，证明运动积分可作为作用量坐标，并据此构造环面真凸 $\mathbb{R}\mathbb{P}^2$ 结构形变空间上辛形式的全局 Darboux 坐标及等距表达式。
In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asymptotic to special Lagrangian cones; hence all type II singularities are modeled by unions of special Lagrangian cones.
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…
We construct a family of compact almost Calabi--Yau manifolds of complex dimension 3 and therein a corresponding family of compact special Lagrangians with one-point singularities modelled upon that T^2-cone constructed by Harvey--Lawson and characterized by Haskins as a stable T^2-cone in the terminology by Joyce.
In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimension. We construct new higher dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfs…
This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-folds in C^3 involving three commuting o.d.e.s. Now special Lagrangian cones in C^3 are linked to the theory of harmonic maps and integrable systems. Harmonic maps from a Riemann surface into complex projective space CP^n are an integrable system, and can be studied and classified…
Roundness of the ample cone and existence of double Lagrangian fibrations on hyperkahler manifolds
本文证明任何满足 $b_2 > 4$ 的最大全纯超凯勒流形存在具有圆凯勒锥和符号 $(1,1)$ 的 Picard 格的形变，由此推出所有已知超凯勒流形具有双横截拉格朗日纤维化的形变，且 Kobayashi 度量在 Picard 秩非最大时消失。
Counting Minimal Lagrangians Via Mirzakhani Functions
研究极小拉格朗日流形的计数问题，证明其数量随面积增长而呈A^{6(k-1)}次方规律，并给出Mirzakhani函数的显式主系数。
We show that any closed oriented immersed Hamiltonian stationary isotropic surface $Σ$ with genus $g_Σ$ in $S^{5}\subset\mathbb{C}^{3}$ is (1) Legendrian and minimal if $g_Σ=0$; (2) either Legendrian or with exactly $2g_Σ-2$ Legendrian points if $g_Σ\geq1.$ In general, every compact oriented immersed isotropic submanifold $L^{n-1}\subset S^{2n-1}\subset\mathbb{C}^{n}$ such that the cone $C\left( L^{n-1}\right) $ is Hamiltonian stationary must be Legendrian and minimal if its first Betti number is zero. Corresponding results for non-orientable links are also provided.
This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct, G-invariant special Lagrangian cones on T^{m-1} in C^m. These examples are interesting as local models for singularities of special Lagrangian subm…
Strict stability of calibrated cones
该研究针对带孤立奇点的校准锥严格稳定性问题，证明特殊拉格朗日锥与余结合锥的严格稳定性，并给出复情形下非严格稳定的实例。
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。
This paper is a follow-up to an earlier paper math.DG/0410260 on desingularizations of Calabi-Yau 3-folds with a conical singularity. In math.DG/0410260 we study Calabi-Yau 3-folds M_0 with a conical singularity at x modelled on some Calabi-Yau cone V, and construct a desingularization of M_0 by gluing in an Asymptotically Conical (AC) Calabi-Yau 3-fold Y to M_0 at x. In this paper, we shall investigate a similar desingularization problem on special Lagrangian (SL) 3-folds in the corresponding Calabi-Yau 3-folds. More precisely, suppose M_0 is now a Calabi-Yau 3-fold with finitely many conical…
A surface $Σ\subset S^5 \subset \mathbb{C}^3$ is called \emph{special Legendrian} if the cone $0 \times Σ\subset \mathbb{C}^3$ is special Lagrangian. The purpose of this paper is to propose a general method toward constructing compact special Legendrian surfaces of high genus. It is proved \emph{there exists a compact, orientable, Hamiltonian stationary Lagrangian surface of genus $1+\frac{k(k-3)}{2}$ in $ \mathbb{C}P^2$ for each integer $ k \geq 3$, which is a smooth branched surface except at most finitely many conical singularities.} If this surface is smooth, it is minimal and the Legendri…
By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elon…
This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds in C^3, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko and Bryant. An important class of ruled SL 3-folds is the special Lagrangian cones in C^3. Each ruled SL 3-fold is asymptotic to a unique SL cone. …
The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the Geometric Langlands program, due to Beilinson-Drinfeld, since it insures that the D-modules on the moduli space of G-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic, hence, e.g. have finite length.
In this paper, which is a natural continuation of our previous paper math.DG/0504557, we describe some special Lagrangians of cohomogeneity one in the resolved conifold. Our main result gives a foliation of the resolved conifold by T^2-invariant special Lagrangians, where the generic leaf is topologically T^2 X R. We also obtain a family of SO(3)-invariant special Lagrangians. These special Lagrangian families in both the deformed and the resolved conifold approach asymptotically the same special Lagrangian cones in the conifold.
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 for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curve) data.
This is the first in a series of five papers math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272, which surveys the series, gives examples, and applies the results to prove some conjectures. This first paper lays the foundations for the series, giving definitions and proving auxiliary results in symp…
This is the last in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with this paper. We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the boundary of a moduli space of compact, nonsingular SL m-folds N in M. We prove the…
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