Monotonicity conjecture for extremal Lagrangian tori in complex projective space

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The monotonicity conjecture for extremal Lagrangian tori in complex projective space

   Let $L$ be a Lagrangian torus in $\mathbb{C}\mathbb{P}^n$. Call $L$ extremal if $A_{\rm \min}(L)=c_L(\mathbb{C}\mathbb{P}^n)$, and call it monotone if its Maslov class is positively proportional to its symplectic area class on $\pi_2(\mathbb{C}\mathbb{P}^n,L)$, namely $\mu=2a[\omega]$ for some $a>0$. The extremal-torus monotonicity conjecture. Every extremal Lagrangian torus in $\mathbb{C}\mathbb{P}^n$ is monotone. The preceding result proves the converse implication: every monotone Lagrangian torus is extremal. The stated implication remains open in the supplied text.

   source: Kai Cieliebak and Klaus Mohnke, “Punctured holomorphic curves and Lagrangian embeddings”, arXiv:1411.1870 (2014).