Some estimates related to Oh's conjecture for the Clifford tori in CP^n

TL;DR

This paper establishes explicit lower bounds on the volume of Lagrangian tori in the Hamiltonian class of Clifford tori in complex projective space, supporting Oh's conjecture with new quantitative estimates.

Contribution

It provides the first explicit constants for volume lower bounds of Hamiltonian deformations of Clifford tori, using Floer homology and integral geometry techniques.

Findings

Explicit lower bounds on volume for Lagrangian tori in Hamiltonian class.

The best known lower bound for the 2-dimensional case is 3/π.

The method combines Floer homology and integral geometry to derive estimates.

Abstract

This note is motivated by Y.G. Oh's conjecture that the Clifford torus $L_n$ in $\mathbb{C}P^n$ minimizes volume in its Hamiltonian deformation class. We show that there exist explicit positive constants $a_n$ depending on the dimension with $a_2=3/\pi$ such that for any Lagrangian torus $L$ in the Hamiltonian class of $L_n$ we have $vol(L) \geq a_n vol (L_n)$. The proof uses the recent work of C.H. Cho on Floer homology of the Clifford tori. A formula from integral geometry enables us to derive the estimate. We wish to point out that a general lower bound on the volume of $L$ exists from the work of C. Viterbo. Our lower bound $a_2= 3/\pi$ is the best one we know.