Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

Abstract

Let Σ be a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold (M, ω). We consider the evolution of Σ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms ω ′ and ω ′′ that determine different orientations and Σ is symplectic with respect to both ω ′ and ω ′′ , we prove the mean curvature flow of Σ exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

“…For further details see also Wang [17,Section 7]. Lemma 2.1 For an arbitrary immersion f : Σ n → R m the second fundamental form satisfies…”

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

“…Proof We will give only a sketch of the proof, since the argument is similar to that of Proposition 2.1 in [Wa1]. Calculate…”

“…But from the monotonicity formula above we see (cf. the argument in [Wa1]) that the blow-up limit satisfies H = 0 and f ⊥ = 0 and hence is a plane. This is a contradiction.…”

“…By analogy with the mean curvature flow [Wa1] we might expect that along the Special H-flow, Type I singularities do not form. A Type I singularity for the H-flow at a time T is a singularity satisfying the estimate…”

“…Motivated by a question of Yau on how to deform a symplectic submanifold into a holomorphic one, Wang [Wa1] showed that if the initial submanifold is symplectic, it remains so along the mean curvature flow and that Type I singularities do not form (see also [ChLi1]). Convergence results have been obtained in a number of special cases [Wa1,Wa2], [TsWa], [ChLiTi]. Also, Lagrangian submanifolds remain Lagrangian along the mean curvature flow [Sm1].…”