A mean-curvature flow along a Kähler–Ricci flow

Let [Formula: see text] be a Kähler surface, and [Formula: see text] an immersed surface in [Formula: see text]. The Kähler angle of [Formula: see text] in [Formula: see text] is introduced by Chern and Wolfson [Am. J. Math. 105 (1983) 59–83]. Let [Formula: see text] evolve along the Kähler–Ricci flow, and [Formula: see text] in [Formula: see text] evolve along the mean-curvature flow. We show that the Kähler angle [Formula: see text] satisfies the evolution equation [Formula: see text] where [Formula: see text] is the scalar curvature of [Formula: see text]. The equation implies that if the initial surface is symplectic (Lagrangian), then, along the flow, [Formula: see text] is always symplectic (Lagrangian) at each time [Formula: see text], which we call a symplectic (Lagrangian) Kähler–Ricci mean-curvature flow. In this paper, we mainly study the symplectic Kähler–Ricci mean-curvature flow.

TRANSLATING SOLITONS TO SYMPLECTIC AND LAGRANGIAN MEAN CURVATURE FLOWS

In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the Kähler angle α of a symplectic translating soliton with max |A| = 1 satisfies that [Formula: see text] where T is the direction in which the surface translates.

Minimal surfaces in ℂPn with constant curvature and Kähler angle

In [2] we used the theory of harmonic sequences to obtain a congruence theorem for minimal immersions of surfaces into ℂPn. In this paper we show how these ideas may be used to give a simplified and unified treatment of some results of Kenmotsu [9, 10], Bryant [4] and Ohnita [13].