In this project we study the energy functional on the set of Lagrangian tori in CP2.
The energy functional has been introduced in [2] as integral of the potential of 2D periodic
Schrödinger operator associated to Lagrangian torus. It has been conjectured in [2] that
the Clifford torus is the unique global minimum of energy functional (the statement is later
referred to as the energy conjecture). Due to geometric interpretation of energy functional as
linear combination of the volume andWillmore functionals, this conjecture can be seen as
the CP2 analogue of the well-known Willmore conjecture for tori in R3, recently proved in
[18].
The energy conjecture has been verified for two families of Hamiltonian-minimal Lagrangian
tori in [2]. Results of [5] and [23] imply the conjecture for minimal Lagrangian tori
of sufficiently high spectral genus and non-embedded minimal Lagrangian tori, respectively.
In the present work we prove the energy conjecture for a family of Hamiltonian-minimal
Lagrangian tori in CP2 constructed in [4]. In sharp distinction with cases considered in [2],
the value of the energy functional for these tori can not be calculated exactly. The proof relies
on analytic bounds for certain elliptic integrals arising from the induced metric of tori.
Possible directions of further work are:
1. Consider local behaviour of the energy functional. Are the critical points of the energy
functional governed by an integrable PDE, akin to Tzizeica equation describing minimal
Lagrangian tori? The same questions for critical points under Hamiltonian variations.
2. Is there an analogue of the energy conjecture for other Kähler-Einstein surfaces? The
case of K3 surface is of special interest as minimal Lagrangian tori in K3 can be related
to elliptic fibrations (for instance [20]) making the conjecture amenable to algebrogeometric
analysis.
3. Examples of monotone Lagrangian tori with trivial Floer cohomology were constructed
in [21]. Do there exist critical points of the energy functional with trivial Floer cohomology?