Angle theorems for the Lagrangian mean curvature flow

Abstract

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle α for the corresponding Lagrangian submanifold in the cross product space L × M satisfies osc(α) ≤ π. If one considers a 4-dimensional Kähler-Einstein manifold M of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that L ⊂ M is a compact oriented Lagrangian submanifold w.r.t. J such that the Kähler form κ w.r.t. K restricted to L is positive and osc(α) ≤ π, then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. κ.

Translating Solitons to Symplectic and Lagrangian Mean Curvature Flows

“…Suppose Σ is Lagrangian and evolves by the mean curvature, Smoczyk has shown that ( [19], [20], [21]),…”

A convergence result of the Lagrangian mean curvature flow

“…We remark that both equations (2.1) and (3.1) are derived in Lemma 5.3 of [10] where p in [10] and η are related by η 2 = 4 p and S = 2c. We claim the following differential inequality is true:…”

“…Denote by Σ t the time slice of the flow at t. That Σ t remains a Lagrangian submanifold follows from a result of Smoczyk [9]. The long-time existence and convergence problems of this flow were studied in [10] and [12].…”

“…However, only C 0 convergence was achieved in the case when c = −1 or 0. Independently, in [10], Smoczyk studied the case when c = −1 or 0 assuming an extra angle condition. He discovered a curvature estimate and showed that the second fundamental form is uniformly bounded under this condition, and thus established the long time existence and C ∞ convergence at infinity.…”

On Donaldson’s flow of surfaces in a hyperkähler four-manifold

“…Also, Lagrangian submanifolds remain Lagrangian along the mean curvature flow [Sm1]. This case has been of great interest in light of the SYZ conjecture [StYaZa] in Mirror Symmetry [ThYa] and there are a number of promising results [Sm2,Sm3], [SmWa], [Wa3], [ChLi2]. However, the Lagrangian mean curvature flow can exhibit some very bad behavior in general (see e.g.…”