Singularities of Lagrangian Mean Curvature Flow

Abstract

Lagrangian mean curvature flow is a promising tool in the study of special Lagrangians. Though the flow has been of interest now for several decades, singularities of the flow are still not well understood, and extensions and generalisations of the flow are still being uncovered and explored. The work of this thesis investigates two distinct topics, both of which shed light on the structure and behaviour of Lagrangian mean curvature flow. We first demonstrate the existence of a boundary condition for Lagrangian mean curvature flow in Calabi-Yau manifolds which preserves the Lagrangian condition. The boundary condition is a generalisation of the constant Lagrangian angle difference between intersecting special Lagrangian submanifolds. This work applies and extends the original work of Smoczyk [60] which proves that the class of closed Lagrangian submanifolds in Calabi-Yau manifolds is preserved. We also investigate singularities of equivariant and almost-calibrated Lagrangian mean curvature flow – flows in Cn with an O(n)-symmetry and a pinching condition on the Lagrangian angle. Given a singularity, we prove that any Type I blowup is a unique pair of planes P1 + P2, any Type II blowup is the Lawlor neck SLaw with asymptotes P1 + P2, and any ‘intermediate’ blowup is P1 + P2. We also prove conditions for longtime existence and singularity formation of the flow. Finally, we investigate the relationship between these topics. We prove that any almost-calibrated equivariant Lagrangian mean curvature flow with boundary on the Lawlor neck converges in infinite time to a special Lagrangian disc, and that the same is true for any rescaled Lagrangian mean curvature flow with boundary on the Clifford torus (assuming extra conditions on the Lagrangian angle).

Date of Award	28 Sept 2020
Original language	English
Awarding Institution	University College London
Supervisor	Felix Schulze (Supervisor)