Energy functional for Lagrangian tori in $\mathbb{C}P^2$

TL;DR

This paper introduces an energy functional for Lagrangian tori in complex projective space, explores its properties, and conjectures the Clifford torus minimizes this energy, with implications for deformations preserving area.

Contribution

It defines a new energy functional for Lagrangian tori in ${f C}P^2$ and investigates its minimizers and deformation properties, linking geometry and integrable systems.

Findings

The energy functional has a geometric interpretation related to the Schrödinger potential.

The Clifford torus is conjectured to minimize the energy functional.

Deformations preserving conformal type also preserve the area of minimal Lagrangian tori.

Abstract

In this paper we study Lagrangian tori in ${\mathbb C}P^2$. A two-dimensional periodic Schr\"odinger operator is associated with every Lagrangian torus in ${\mathbb C}P^2$. We introduce an energy functional for tori as an integral of the potential of the Schr\"odinger operators, which has a natural geometrical meaning. We study the energy functional on two families of Lagrangian tori and propose a conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori. In particular we show that if the deformation preserves a conformal type of the torus, then it also preserves the area of the torus. Thus it follows that deformations generated by Novikov-Veselov equations preserve the area of minimal Lagrangian tori.