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Curvature inequalities for Lagrangian submanifolds: the final solution

Abstract : Let M-n be an n-dimensional Lagrangian submanifold of a complex space form (M-n) over tilde (4c) of constant holomorphic sectional curvature 4c. We prove a pointwise inequality delta(n(1),...,n(k)) <= a(n,k,n(1),...n(h))parallel to H parallel to(2) + b(n,k,n(1),...n(k))c, with on the left-hand side any delta-invariant of the Riemannian manifold M-n and on the right-hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right-hand side are optimal in the sense that there exist non-minimal examples satisfying equality at least one point. We also characterize those Lagrangian submanifolds satisfying equality at any of their points. Our results correct and extend those given in [6]
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Submitted on : Tuesday, May 25, 2021 - 10:28:36 AM
Last modification on : Friday, March 11, 2022 - 3:53:10 PM

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B.-Y. Chen, F. Dillen, J. van Der Veken, Luc Vrancken. Curvature inequalities for Lagrangian submanifolds: the final solution. Differential Geometry and its Applications, Elsevier, 2013, 31 (6), pp.808-819. ⟨10.1016/j.difgeo.2013.09.006⟩. ⟨hal-03233905⟩

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