Second order families of special Lagrangian 3-folds
Robert L. Bryant
备注：
36 pages, latex2e, some extra comments included in response to reader questions, many typos corrected, and important references added
发表信息：
Perspectives in Riemannian Geometry, CRM Proceedings and Lecture Notes, edited by Vestislav Apostolov, Andrew Dancer, Nigel Hitchin, and McKenzie Wang, vol. 40 (2006), American Mathematical Society
A second order family of special Lagrangian submanifolds of complex m-space is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of complex 3-space is characterized by a single algebraic equation on the second fundamental form.
While the `generic' set of such conditions turns out to be incompatible, i.e., there are no special Lagrangian submanifolds that satisfy them, there are many interesting sets of conditions for which the corresponding family is unexpectedly large. In some cases, these geometrically defined families can be described explicitly, leading to new examples of special Lagrangian submanifolds. In other cases, these conditions characterize already known families in a new way. For example, the examples of Lawlor-Harvey constructed for the solution of the angle conjecture and recently generalized by Joyce turn out to be a natural and easily described second order family.
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.
Constructing special Lagrangian m-folds in C^m by evolving quadrics
这是第二次在一系列文件建设明确的例子特殊拉格朗日子流形在C^m。第一篇论文是math.DG/0008021，研究了具有大对称群的特殊拉格朗日m-折叠。第三个是数学。DG/0010036，它使用本文的思想来构造C^3中的特殊拉格朗日3-折叠族。本文给出了C ^m中由Lagrange平面R^m中的二次曲面（m-1）-子流形所包围的特殊Lagrange m-折叠的构造。一般来说，它们只有离散的对称群。我们的一些例子已经由Lawlor和Harvey使用不同的方法构建。这些文件的主要动机是奠定了基础，研究的奇异性紧凑的特殊拉格朗日m-褶皱的Calabi-Yau m-褶皱。理解这种奇异性对于解决关于卡-丘3-折叠镜像对称的SYZ猜想具有重要意义。本文构造的C^m中的特殊Lagrange m-褶皱包含了S^a x S^b x S ^1（a+b=m-2）上的许多锥，它们是Calabi-Yau m-褶皱中特殊Lagrange m-褶皱奇点的局部模型.
Evolution equations for special Lagrangian 3-folds in C^3
这是第三次在一系列文件建设明确的例子特殊拉格朗日子流形在C^m。该系列的前一篇论文math.DG/0008155定义了演化数据的概念，其中包括R^n中的（m-1）-子流形P，并构造了C^m中的一类特殊拉格朗日m-折叠N，它们被P在满足一阶o.d.e.的线性或仿射映射phi_t：R^n -> C^m的单参数族下的像扫出。于t.在本文中，我们使用同样的思想来构造C^3中的特殊拉格朗日3-折叠。我们发现m=3的演化数据集与齐次辛2-流形P之间存在1-1对应关系，这使我们能够写出几组有趣的演化数据集，从而构造C^3中相应的特殊拉格朗日3-折叠族。我们的主要结果是C^3中的一些新的特殊拉格朗日3-折叠族，我们用参数形式非常明确地写出来。一般地，这些是非奇异的浸入3-子流形，并且同构于R^3或S^1 x R^2。一些3倍是奇异的，我们描述他们的奇异性，我们相信这是一种新的。我们希望这些3-折叠将有助于理解卡-丘3-折叠中紧致特殊拉格朗日3-折叠的奇异性。这对解决镜像对称中的SYZ猜想有重要意义。
Second Order Families of Special Lagrangian Submanifolds in C^4
本文将Robert Bryant的文章“Second Order Families of Special Lagrangian 3-folds”中的结果推广到4维。我们考虑C^4中特殊的拉格朗日4重的分类问题，其基本三次函数在SO（4）中的每一点都有一个非平凡的稳定子。SO（4）-稳定子非平凡的特殊拉格朗日4重上的点类似于经典曲面理论中的脐点。在证明特殊Lagrange 4-折叠族的存在性时，我们使用了Cartan-Kahler理论中的外微分系统方法。这种方法保证告诉我们是否存在任何一族具有某种对称性的特殊拉格朗日子流形，但没有给出子流形的显式描述。为了得到一个明确的描述，我们研究了子流形的叶理和其他几何特殊性。用这种方式，我们解决了许多情况，并以明确的方式描述了特殊的拉格朗日子流形的家庭。
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.
Immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and its deviation from Joyce's criteria: Potential image-support rigidity of A-branes that wrap around a sL $S^3$
The number of closed billiard trajectories in a rational-angled polygon grows quadratically in the length. This paper gives an analogue on K3 surfaces, by considering special Lagrangian tori. The analogue of the angle of a billiard trajectory is a point on a twistor sphere, and the number of directions admitting a special Lagrangian torus fibration with volume bounded by $ V $ grows like $ V^{20} $ with a power-saving term. Bergeron--Matheus have explicitly estimated the exponent of the error term as $ {20-\frac{4}{697633} }$. The counting result on K3 surfaces is deduced from a count of primi…
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.
We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subsets of the Kähler cone. Using the same technique, we show that the Kähler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.
We obtain families of non-isotopic closed exact Lagrangian submanifolds in quasi-projective holomorphic symplectic manifolds that admit contracting $\mathbb{C}^*$-actions. We show that the Floer cohomologies of these Lagrangians are topological in nature, recovering the ordinary cohomologies of their intersection. Moreover, by using these Lagrangians and a version of Carrell-Goresky's integral decomposition theorem, we obtain degree-wise lower bounds on the symplectic cohomology of these spaces.
