§ 5.42 Finite-time Lagrangian neck pinches with circle symmetry06177EF7798F

In an ALE or ALF hyperkähler four-manifold with a tri-Hamiltonian circle action, Lotay–Oliveira analyze the flow of embedded almost-calibrated circle-invariant Lagrangian surfaces. For compact surfaces, or the specified noncompact surfaces asymptotic to a pair of planes, finite-time singularities are neck pinches with the local models established in Theorem 1.1. Their examples guarantee such singularities under the geometric conditions of Section 4. Noncompact examples exist in the nontrivial Gibbons–Hawking settings described there; compact examples require the appropriate configuration of special Lagrangian spheres and phases. This is not a claim for every ambient manifold or every initial Lagrangian. For compact connected initial data, the flow can be continued through finitely many such singular times and converges to an $A_ k$-chain of embedded special Lagrangian spheres, with the precise current and phase conclusions of Theorem 1.4. A single smooth parameter manifold is not retained through a neck pinch. In the Gibbons–Hawking base, the surface is encoded by an arc and circle fibers; the evolution is a weighted curve-shortening equation. The diagram shows the arc approaching a center, the collapsing fiber and the separated chain after the pinch. It is a schematic of the proved mechanism, not a Euclidean surface of revolution asserted to solve the hyperkähler flow.

Sources: [LotayOliveira2023].