ScalingStacks

Example 4.3 . [04D4]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Example 4.3.

Consider a figure eight curve inside ℝ2\mathbb{R}^{2}, 4747 47 Recall that curves in ℝ2\mathbb{R}^{2} are automatically Lagrangian.whose two looms have unequal areas. Along the mean curvature flow (known as the ‘curve shortening flow’ in this context) one loom shrinks first to zero size. At the moment of singularity, the Lagrangian angle at the self intersection point has a jump. From a more generalisable perspective, one notices that each loom encloses a holomorphic disc, and this singularity is associated with one holomoprhic disc shrinking to zero size and disappearing. The general lesson is that the shrinking down of small area holomorphic discs messes up the grading, so it is desirable to exclude them if possible. 4848 48 Indeed, one important ingredient in Neves’s proof of singularity formation [61] is the destruction of grading related to shrinking enclosed 2-dimensional areas. Although it is not explict in Neves’s work, these areas seem related to holomorphic discs.

Original mathematics by the credited authors. Source-backed reader collection; mathematical self-containment is not assessed.