Curvature inequalities for Lagrangian submanifolds: the final solution

TL;DR

This paper establishes optimal pointwise inequalities relating delta-invariants, mean curvature, and holomorphic sectional curvature for Lagrangian submanifolds in complex space forms, extending and correcting previous results.

Contribution

It proves a new class of optimal inequalities for Lagrangian submanifolds and characterizes those satisfying equality, advancing the understanding of their geometric properties.

Findings

Derived a pointwise inequality involving delta-invariants, mean curvature, and curvature.

Identified conditions for equality and characterized submanifolds satisfying them.

Extended and corrected previous inequalities in the literature.

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].