Special Lagrangians of cohomogeneity one in the resolved conifold

Special Lagrangian conifolds, I: Moduli spaces

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…

Geometric transitions with Spin(7) holonomy via a dynamical system

We clarify the global geometry of two 1-parameter families of cohomogeneity one Spin(7) holonomy metrics with generic orbit the Aloff--Wallach space $N(1,-1) \cong \mathrm{SU}(3)/\mathrm{U}(1)$ and singular orbits $S^5$ and $\mathbb{C}P^2$, which at short distance were shown to exist by Reidegeld. The two families fit into the geography of previously known families of cohomogeneity one metrics with exceptional holonomy and provide a Spin(7) analogue of the well-known conifold transition in the setting of Calabi--Yau 3-folds. Furthermore, we discover that there is another transition to families…

具有对称性的 $\mathbb{C}^m$ 中的拉格朗日平移孤立子与特殊拉格朗日子流形

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。

Special Lagrangian conifolds, II: Gluing constructions in C^m

We prove two gluing theorems for 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. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the first (non-trivial) examples of SL conifolds which have a conical singularity but are not, globally, cones. We also obtain: (i) a desingularization procedure for transverse intersection and self-intersection points, using "Lawlor necks"; (ii) a construction which completely desi…

具有定量精确动力学的拉格朗日平均曲率流的有限时间奇点

Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

构建几乎校准的拉格朗日平均曲率流，从接近特殊拉格朗日的初值出发，在有限时间产生II型奇点，给出曲率爆破率，奇点切流为特殊拉格朗日锥，爆破极限是光滑特殊拉格朗日去奇异化。

关于在欧几里得空间中具有共效性的子流形的一些结果

Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two

本文研究了在欧几里得空间中具有共效性的子流形，通过构造特殊Lagrangian子流形并证明其刚性结果。

Conormal bundles to knots and the Gopakumar-Vafa conjecture

We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in …

特殊拉格朗日圈与卡拉比-丘转移

Special Lagrangian cycles and Calabi-Yau transitions

研究在Fu-Li-Yau几何的非Kähler紧三流形中构造特殊拉格朗日3-球，揭示其源于Calabi-Yau拓扑转移，且conifold转移将全纯2-圈替换为特殊拉格朗日3-圈。