Juergen Jost, Yuan-Long Xin
We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.
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We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.
The aim of this expository article is to reveal the equivalence of Jörgens' Theorem on the two dimensional unimodular Hessian equation and Fu's Theorem on the two dimensional special Lagrangian equation.
We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.
In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$
Let Σbe a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σlies in one of these regions, then Σis an affine space.
Bernstein theorems for nonlinear geometric PDEs
本文围绕极小曲面、Monge–Ampère、特殊 Lagrangian 方程及极小曲面系统，综述几何偏微分方程中的 Bernstein 问题，内容源自 2024 年欧洲微分几何博士学校讲座。
We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth special Lagrangians. We also use soft methods from symplectic geometry (the relative version of the h--principle) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singulariti…
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.
The purpose of this paper is to give an application of the gluing theorem for special Lagrangian submanifolds of a Calabi-Yau 3-fold. We proved a gluing theorem before to smooth a codimension-two singularity of a particular special Lagrangian submanifold. In this paper we will show that this theorem can be applied to more general cases where different special Lagrangians are intersecting and gives a way of constructing new special Lagrangian submanifolds. As an example we will show that a smooth special Lagrangian submanifold can be obtained from five copies of RP^3 intersecting pairwise in a …
The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of immersed special Lagrangian submanifolds $\tilde{L}_t$, that are diffeomorphic to a branched covering of $L$ and $\tilde{L}_t$ convergence to $2L$ as current. This answers a question suggested by Donaldson. Combining with the work of Abouzaid and Imagi, we obtain constraints for the existence of nondegenerate $\mathbb{Z}_2$ harmonic 1-forms.
In this article we obtain a classification of special Lagrangian submanifolds in complex space forms subject to an $SO(2)\rtimes S_3$-symmetry on the second fundamental form. The algebraic structure of this form has been obtained by Marianty Ionel. However, the classification of special Lagrangian submanifolds in $\mathbb{C}^4$ having this $SO(2)\rtimes S_3$ symmetry in that paper is incomplete. In the present paper we give a complete classification of such submanifolds, and extend the classification to special Lagrangian submanifolds of arbitrary complex space forms with $SO(2)\rtimes S_3$-sy…
The purpose of this paper is to prove a gluing theorem for a given special Lagrangian submanifold of a Calabi-Yau 3-fold. The proof will be an adaption of the gluing techniques in J-holomorphic curve theory. It is a well known procedure in geometric analysis to construct new solutions to a given nonlinear partial differential equation by gluing known solutions. First an approximate solution is constructed and then using analytic methods it is perturbed to a real solution. In this paper the gluing theorem will be used for smoothing a singularity of a special Lagrangian submanifold. In particula…
We define the notion of special Lagrangian curvature, showing how it may be interpreted as an alternative higher dimensional generalisation of two dimensional Gaussian curvature. We obtain first a local rigidity result for this curvature when the ambiant manifold has negative sectional curvature. We then show how this curvature relates to the canonical special Legendrian structure of spherical subbundles of the tangent bundle of the ambiant manifold. This allows us to establish a strong compactness result. In the case where the special Lagrangian angle equals $(n-1)π/2$, we obtain compactness …
Transversality for perturbed special Lagrangian submanifolds
本文在配备广义Calabi-Yau结构的6维流形上，证明了扰动特殊拉格朗日子流形模空间的横截性定理，并表明该模空间一般由孤立点组成，为Donaldson-Segal计数方案提供了关键理论支撑。
We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian-Yau. We prove that if $X$ is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then $X$ admits infinitely many disjoint special Lagrangians. In complex dimension $2$, we prove that if $Y$ is a del Pezzo surface, or a rational elliptic surface, and $D\in |-K_{Y}|$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ admits a special Lagrangian torus fibration, as conjectured by …
R.C.McLean showed that the moduli space of nearby submanifolds of a smooth, compact, orientable special Lagrangian submanifold L in a Calabi-Yau manifold X is a smooth manifold and its tangent space at L is identified with the space of harmonic one forms on L. In this paper, we will extend this result from Calabi-Yau manifolds to the symplectic manifolds with non-integrable almost complex structure.
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…
We prove two main results: (a) Suppose $L$ is a closed, embedded, exact special Lagrangian $m$-fold in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse special Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $L$ is one of the explicit 'Lawlor neck' family of examples found by Lawlor (Invent. math. 95, 1989). (b) Suppose $L$ is a closed, embedded, exact Lagrangian mean curvature flow expander in ${\mathbb C}^m$ for $m\ge 3$ asymptotic at infinity to the union $Π_1\cupΠ_2$ of two transverse Lagrangian planes $Π_1,Π_2$ in ${\mathbb C}^m$. Then $…
We discuss the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotically conical, ends. Our main theorem is the natural next step in the chain of results initiated by McLean and continued by the author and by Joyce. We emphasize a unifying framework for studying the various cases and discuss analogies and differences between them. This paper also…
The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkaehler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on long-time existence of the Lagrangian mean curvature flow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.
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。
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.
We study gluings of asymptotically cylindrical special Lagrangian submanifolds in asymptotically cylindrical Calabi--Yau manifolds. We prove both that there is a well-defined gluing map, and, after reviewing the deformation theory for special Lagrangians, prove that this gluing map defines a local diffeomorphism from matching pairs of deformations of asymptotically cylindrical special Lagrangians to deformations of special Lagrangians. We also give some examples of asymptotically cylindrical special Lagrangian submanifolds to which these results apply.
We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\t…
Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimension 3.
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.
We give a necessary and sufficient condition for a special Lagrangian submanifold in C^n constructed by Lawlor being also special Lagrangian in C^n with the Fubini-Study form.
Moduli of special Lagrangians with boundary, II: Lagrangian Flux and Affine Structures
本文延续对卡拉比-丘流形中带边特殊拉格朗日子流形模空间的研究，在其上构造了一对特殊仿射结构与一个黑塞度量，揭示了该模空间的几何结构。
In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.
Geodesics of positive Lagrangians from special Lagrangians with boundary
研究正拉格朗日子流形空间中的测地线，通过柱形变换揭示其与特殊拉格朗日柱体的关系，并证明相关空间的开性，提供非对称维度下的测地线解例。
We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi-Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey-Lawson, in the Ri…
We construct smooth families of compact special Lagrangian submanifolds embedded in some toric hyper-K\"ahler manifolds, which never become holomorphic Lagrangian submanifolds via any hyper-Kähler rotations. These families converge to special Lagrangian immersions with self-intersection points in the sense of current. To construct them, we apply the desingularization method developed by Joyce.
In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed conifold by T^2-invariant special Lagrangian 3-folds, where the generic leaf is topologically T^2 X R. In the limit, these special Lagrangians asymptotically approach a special Lagrangian cone on a torus in the conifold. Using moment map techniques we also recover the family of SO(n)-invariant special Lagrangian n-folds in the cotangent bundle of the n-sphere obta…
Moduli of special Lagrangians with boundary, I: Unobstructed Deformations
本文研究卡拉比-丘流形中边界约束在给定拉格朗日子流形上的带边特殊拉格朗日子流形的形变问题，证明其形变无阻碍，且模空间为光滑流形，维数等于一阶相对上同调维数。
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…
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.
In this paper, the relationship between the existence of special lagrangian submanifolds and the collapsing of Calabi-Yau manifolds is studied. First, special lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi-Yau manifolds. Then, in the opposite direction,it is shown that the existence of special lagrangian submanifolds with small volume implies the collapsing of some regions in the ambient Calabi-Yau manifolds.
In this paper we construct all smooth torus fibres of the generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.
We derive formulas for the mean curvature of special Lagrangian 3-folds in the general case where the ambient 6-manifold has intrinsic torsion. Consequently, we are able to characterize those SU(3)-structures for which every special Lagrangian 3-fold is a minimal submanifold. In the process, we obtain an obstruction to the local existence of special Lagrangian 3-folds.
This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$, viewed as a nearly Kähler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{SU}(2)$ moment-type maps. We describe new homogeneous examples, from both perspectives, and classify totally geodesic special Lagrangian submanifolds. We show that every special Lagrangian in $\mathbb{CP}^3$, or the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$ admitting a symmetry of an $\mathrm{SU}(2)$ subgroup of nearly Kähler automorphisms is automatically homogeneous.
We introduce a notion of vanishing Maslov index for lagrangian varifolds and lagrangian integral cycles in a Calabi-Yau manifold. We construct mass-decreasing flows of lagrangian varifolds and lagrangian cycles which satisfy this condition. The flow of cycles converges, at infinite time, to a sum of special lagrangian cycles (possibly with differing phases). We use the flow of cycles to obtain the fact that special lagrangian cycles generate the part of the lagrangian homology which lies in the image of the Hurewicz homomorphism. We also establish a weak version of a conjecture of Thomas-Yau r…
We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-Hölder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.
This paper extends to dimension 4 the results in the article "Second Order Families of Special Lagrangian 3-folds" by Robert Bryant. We consider the problem of classifying the special Lagrangian 4-folds in C^4 whose fundamental cubic at each point has a nontrivial stabilizer in SO(4). Points on special Lagrangian 4-folds where the SO(4)-stabilizer is nontrivial are the analogs of the umbilical points in the classical theory of surfaces. In proving existence for the families of special Lagrangian 4-folds, we used the method of exterior differential systems in Cartan-Kahler theory. This method i…
Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,ω).$ We define a functional $\CC:\OO \to \R$ for each differential form $β$ of middle degree satisfying $β\wedge ω= 0$ and an exactness condition. If the exactness condition does not hold, $\CC$ is defined on the universal cover of $\OO.$ A particular instance of $\CC$ recovers the Calabi homomorphism. If $β$ is the imaginary part of a holomorphic volume form, the critical points of $\CC$ are special Lagrangian submanifolds. We present evidence that $…
We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi.
I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals. This method of constructing special Lagrangian submanifolds is well known. What does not appear to be in the current literature is an explicit description of such examples in which the special Lagrangian submanifold is a 3-torus.
This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lagrangian 3-folds in C^3. This paper describes a construction of special Lagrangian m-folds in C^m which are fibred by (m-1)-submanifolds which are quadrics in Lagrangian planes R^m in C^m. Generically they have only discrete symmetry groups. Some of our examples have been previous…
In this paper we construct monodromy representing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.
We show a method to construct a special Lagrangian submanifold L' from a given special Lagrangian submanifold L in a Calabi-Yau manifold with the use of generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. By our method, we construct some non-trivial examples in the cotangent bundles of the spheres which are non-flat Calabi-Yau manifolds equipped with the Stenzel metrics.
Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they deviate from being deformable to a family of special Lagrangian deformations by examining in detail Joyce's criteria on this family. The result suggests a potential image-support rigidity of A-branes that wrap around a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which res…
From tropical curves to special Lagrangians
该研究通过粘合构造匹配特殊拉格朗日局部模型与热带曲线的组合结构，证明单位边权的局部平面热带曲线可实现为$T^*T^n$中特殊拉格朗日子流形族的缩放矩映射像极限，建立了两类几何的直接联系。
We construct new special Lagrangian submanifolds in complex Euclidean space using a pair of minimal Legendrian submanifolds in odd-dimensional spheres and certain Lagrangian surface belonging to a family that can be considered as a generalization of the special Lagrangian surfaces in complex Euclidean plane. Our examples include those invariant under the standard action of SO(p+1)xSO(q+1) on C^n = C^(p+1) x C^(q+1), n=p+q+2.
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 $m\ne 8,9$.
Special Lagrangian submanifolds and circle collapse on K3
研究K3曲面塌缩到三维仿射基时，仿射线提升为退化特殊拉格朗日球面和环面的序列，并构造连接Taub-NUT气泡对的特殊拉格朗日球面。
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.
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.
This paper gives an example of special Lagrangian manifold obtained from a hypersurface of a complex Grassmannian with vanishing first Chern class. The obtained manifold is a 1-torus bundle over the two dimensional real projective space. Such manifolds are interesting for mirror symmetry theory. Other examples of the same type are provided at the end of this article.
We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject. Special Lagrangian m-folds in C^m are defined, and ways of constructing them described. 'Almost Calabi-Yau manifolds' (a generalization of Calabi-Yau manifolds useful in special Lagrangian geometry) are introduced, and the deformation theory, obstruction theory, and moduli …
Lagrangian Translating Solitons and Special Lagrangians in $\mathbb{C}^m$ with Symmetries
本文在具有对称性的 $\mathbb{C}^m$ 中构造了新的拉格朗日平移孤立子和特殊拉格朗日子流形，推广了 Castro-Lerma 的 Ansatz，并对单群情形给出完整分类，同时描述了关于非紧子群对称的余齐性一例子。
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