abstract:

We study geometric properties of the Lagrangian self-shrinking tori in $\Bbb{R}^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a {\L}ojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with a compactness theorem. When the area bound is small, we show that any Lagrangian self-shrinking torus in $\Bbb{R}^4$ with small area is embedded with uniform curvature estimates, and the space of such tori is compact.

Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in $\Bbb{R}^4$, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in $\Bbb{R}^3$ by Colding-Minicozzi.

摘要：

我们研究了$ \ Bbb {R} ^ 4 $中拉格朗日自收缩托里的几何性质。当面积均匀地定界在上面时，我们证明了拉格朗日自缩环面的熵只能取有限的多个值。这是通过为分支的共形自收缩环导出{\ L} ojasiewicz-Simon型梯度不等式，然后与紧致性定理组合来完成的。当区域边界较小时，我们证明在$ \ Bbb {R} ^ 4 $中具有小面积的任何Lagrangian自收缩环面均以统一的曲率估计值嵌入，并且这种环面的空间紧凑。

利用熵值的有限性，我们为$ \ Bbb {R} ^ 4 $中的拉格朗日沉浸式托里构造了分段拉格朗日平均曲率流，沿此拉格朗日条件得以保留，面积减小，而紧致的I型奇点具有固定区域上限的对象可以有限步地被干扰。这是Colding-Minicozzi在$ \ Bbb {R} ^ 3 $中用于嵌入曲面的构造的拉格朗日版本。