Finite time singularity, and prototypical bad behaviours [04D2]

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

Finite time singularity, and prototypical bad behaviours

The central difficulty of the subject is that finite time singularities are in general inevitable, starting from complex dimension two. Indeed, a theorem of Neves [61, Thm. 6.1] says that for any embedded Lagrangian submanifold inside a Calabi-Yau surface, there exists a Lagrangian within the same Hamiltonian isotopy class, such that the LMCF with this initial data forms finite time singularity. ⁴⁵⁴⁵ 45 Whether the same holds for almost calibrated initial data is an interesting open problem. There is also a good geometric reason why singularities must occur in Joyce’s program: the Thomas-Yau uniqueness theorem applies to Lagrangians within the same derived Fukaya category class, which may include several Hamiltonian isotopy classes, at most one of which can have special Lagrangian representatives. In order for an initial Lagrangian in the wrong Hamiltonian isotopy class to find its way back to the right class along the LMCF, it must undergo a sequence of surgeries.

Now there is a substantial theory of weak solutions of mean curvature flows in the context of varifolds and currents, known as ‘Brakke flows’ [12], which exist under very general conditions. The problem is that such solutions are too weak to guarantee uniqueness of the flow, and the total mass of the varifold may jump down at discrete time. Even more fatally for our purpose, once the smoothness of the flow is dropped, the Lagrangian condition may not be preserved any more. It is instructive to look at the prototypical bad behaviours:

Example 4.2.

Schoen and Wolfson [67] found area minimizers within certain Lagrangian isotopy classes, which are not minimal surfaces.⁴⁶⁴⁶ 46 There is no contradiction: area minimisation among Lagrangians by no means guarantee area stationarity among submanifold. The Brakke flow with such initial data further decreases mass in time, so must cease to be Lagrangian. However, these examples are not graded, so do not contradict Joyce’s program. A possible lesson is that non-graded Lagrangians are bad.

Example 4.3.

Consider a figure eight curve inside $\mathbb{R}^{2}$ , ⁴⁷⁴⁷ 47 Recall that curves in $\mathbb{R}^{2}$ 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.

Example 4.4.

Consider any compact Lagrangian inside the unit ball of $\mathbb{C}^{n}$ . By an easy maximum principle argument, during the flow $L_{t}$ remains inside the shrinking ball $\{\sum|z_{i}|^{2}\leq 1-2nt\}$ , so must develop a finite time singularity at some $t\leq\frac{1}{2n}$ . From the Floer theoretic perspective, since such Lagrangians can always be displaced off itself by the Hamiltonian isotopy corresponding to translations in $\mathbb{C}^{n}$ , its Floer cohomology is either obstructed or zero. As such, a compact Lagrangian supported in a small coordinate ball is invisible to the derived Fukaya category. From a different perspective, since such Lagrangians have zero homology class, they are excluded in the almost calibrated case.