论文标题
同构性 - 一个拉格朗日平均曲率流
Cohomogeneity-One Lagrangian Mean Curvature Flow
论文作者
论文摘要
我们研究了$ \ mathbb {c}^n $中拉格朗日人的平均曲率流，相对于一个紧凑的谎言组$ g \ leq \ leq \ mathrm {su}（n）$在$ \ mathbb {c}^n $上线性地表现。每个这样的拉格朗日都必须在标准矩映射的$μ^{ - 1}（ξ）$的级别中，$μ\ colon \ colon \ mathbb {c}^n \ to \ mathfrak {g}^*$，并且平均曲率曲率保留了此包含。我们对所有同一均匀性进行分类 - 一个自我相似的缩小，扩展和转化为流动的解决方案，以及同时性的同时性 - 一个平滑的特殊拉格朗日人，位于$μ^{ - 1}（0）$中。限制在零级别中几乎被校准的流量设置$μ^{ - 1}（0）$的情况下，我们对有限的奇异性进行了分类，明确描述了I型和II型爆炸模型。最后，鉴于$μ^{ - 1}（0）$中的任何同一性的特殊Lagrangian，我们表明它是Lagrangian MCF Singularity的II型爆炸模型。在整个过程中，我们提供了适当的小组操作的明确示例，包括$ g $简单的完整列表。这产生了许多新的新示例，这些示例是拉格朗日MCF的孤子片缩小和扩展，以及无限的许多新奇异模型。
We study mean curvature flow of Lagrangians in $\mathbb{C}^n$ that are cohomogeneity-one with respect to a compact Lie group $G \leq \mathrm{SU}(n)$ acting linearly on $\mathbb{C}^n$. Each such Lagrangian necessarily lies in a level set $μ^{-1}(ξ)$ of the standard moment map $μ\colon \mathbb{C}^n \to \mathfrak{g}^*$, and mean curvature flow preserves this containment. We classify all cohomogeneity-one self-similarly shrinking, expanding and translating solutions to the flow, as well as cohomogeneity-one smooth special Lagrangians lying in $μ^{-1}(0)$. Restricting to the case of almost-calibrated flows in the zero level set $μ^{-1}(0)$, we classify finite-time singularities, explicitly describing the Type I and Type II blowup models. Finally, given any cohomogeneity-one special Lagrangian in $μ^{-1}(0)$, we show it occurs as the Type II blowup model of a Lagrangian MCF singularity. Throughout, we give explicit examples of suitable group actions, including a complete list in the case of $G$ simple. This yields infinitely many new examples of shrinking and expanding solitons for Lagrangian MCF, as well as infinitely many new singularity models.
