Evans, Christopher G.; (2022) Lagrangian mean curvature flow in the complex projective plane. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In this thesis, we study Lagrangian mean curvature flow of monotone Lagrangians in two different settings, finding interesting and contrasting behaviour in each case. First, we study the self-shrinking Clifford torus in C2. On the one hand, we find a family of Ck-small Hamiltonian deformations that force type II singularities to form. On the other hand, we find that any Hamiltonian deformation restricted to the unit sphere flows back to the self-shrinking Clifford torus after rescaling. Second, we study Lagrangian mean curvature flow in Kähler–Einstein manifolds with positive Einstein constant. We show that monotone Lagrangians do not attain type I singularities under mean curvature flow, an analogue of a result of Wang [49]. Next, we investigate Lagrangian mean curvature flow of Vianna’s exotic monotone tori ([47], [48]) in CP2. We define an (S1 × Z2)-equivariance, and we prove a Thomas–Yau-type result in this setting. We define a surgery procedure and show that any equivariant monotone Lagrangian torus exists for all time under mean curvature flow with surgery, undergoing at most a finite number of surgeries before converging to a minimal Clifford torus. In particular, our result show that there does not exists a minimal equivariant Chekanov torus. Furthermore, we explicitly construct a monotone Clifford torus which has two finite-time singularities under mean curvature flow with surgery, becoming a Chekanov torus before eventually returning to become a Clifford torus again.

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