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In this paper, we investigate the gap phenomena for complete Lagrangian submanifolds satisfying ∇ ∗ T ≡ 0 in complex space forms N n (4 c) , where T = 1 n ∇ ∗ B and B is the Lagrangian trace-free second fundamental form. We prove that under some pointwise or integral curvature pinching condition, the submanifold is either totally geodesic or isometric to the Whitney sphere. [ABSTRACT FROM AUTHOR]

In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi–Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples of Lagrangian mean curvature flows in non-flat Calabi–Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities. [ABSTRACT FROM AUTHOR]

We determine all tight Lagrangian surfaces in S2 × S2. In particular, globally tight Lagrangian surfaces in S2 × S2 are nothing but real forms of this symmetric space. [ABSTRACT FROM AUTHOR]

In this paper we propose an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cn. [ABSTRACT FROM AUTHOR]