The second type singularities of symplectic and lagrangian mean curvature flows

Abstract

This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a Kähler surface. The relation between the maximum of the Kähler angle and the maximum of |H| 2 on the limit flow is studied. The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.

Singularities of symplectic and Lagrangian mean curvature flows

“…The beautiful results on the nature of singularities of the mean curvature flows of convex hypersurfaces have been obtained by Huisken [11], Huisken-Sinestrari [12], [13] and White [24]. In [9] we obtain the relation between the maximum of the Kähler anlge and the maximum of |H| 2 on the blow-up flow of the symplectic mean curvature flow or the calibrated Lagrangian mean curvature flow.…”

“…From the main theorem in [2] and [23], we know that this is a type II singularity. Recall that [9], we can define a sequence of rescaled surfaces Σ k s around (X 0 , T ). For each fixed R > 0, by parabolic estimates, we have that Σ By the evolution equation derived in [2], we see that, along the mean curvature flow Σ ∞ s , cos α satisfies…”

“…The following monotonicity formula for Σ ∞ s is derived in [9]. Let H(X, X 0 , t, t 0 ) be the backward heat kernel on R 4 .…”

“…It is well-known (see [9]) that, a sequence of rescaled surfaces at a singular point (X 0 , T 0 ) converges strongly to a blow-up flow Σ ∞ s , for s ∈ (−∞, 0] if the singular point is of type I, for s ∈ (−∞, +∞) if the singular point is of type II. In this paper, we show that Σ ∞ s is a non trivial holomorphic curve with finite total curvature and bounded Gauss curvature in C 2 , if Σ curvature of a non-flat minimal surface with finite total curvature in C 2 achieves only discrete values −2πN where N is a nature number.…”

“…More precisely, let T be a discrete singular time and (X 0 , T ) be a singular point in M, one shows (see [9]) that there are sequences…”