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Curvature inequalities for Lagrangian submanifolds: the final solution | | Bang-Yen Chen
; Franki Dillen
; Joeri Van der Veken
; Luc Vrancken
; | | Date: |
5 Jul 2013 | | Abstract: | Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form.
We prove a pointwise inequality $$delta(n_1,ldots,n_k) leq
a(n,k,n_1,ldots,n_k) |H|^2 + b(n,k,n_1,ldots,n_k)c,$$ with on the left hand
side any delta-invariant of the Riemannian manifold $M$ 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 sence that there exist
non-minimal examples satisfying equality at 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 [B.-Y. Chen and F.
Dillen, Optimal general inequalities for Lagrangian submanifolds in complex
space forms, J. Math. Anal. Appl. 379 (2011), 229--239]. | | Source: | arXiv, 1307.1497 | | Services: | Forum | Review | PDF | Favorites | |
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