Singularities of codimension two mean curvature flow of symplectic surfaces

TL;DR

This paper analyzes the singularities that occur in the mean curvature flow of compact symplectic surfaces within compact Kaehler-Einstein surfaces, revealing that tangent cones at the first blow-up are unions of complex 2-planes.

Contribution

It establishes that the tangent cone at the first singularity in such flows is a finite union of complex 2-planes, providing new geometric insight into singularity formation.

Findings

Tangent cones at first blow-up are unions of more than two complex 2-planes.

Singularities are characterized by complex geometric structures.

Results apply to compact symplectic surfaces in Kaehler-Einstein manifolds.

Abstract

We prove that for a mean curvature flow of a compact symplectic surface in a compact Kaehler-Einstein surface, the tangent cone at the first blow-up time consists of a finite union of more than two 2-planes in $R^4$ which are complex in a complex structure on $R^4$.