On Hamiltonian stationary Lagrangian spheres in non-Einstein Kaehler surfaces

TL;DR

This paper investigates Hamiltonian stationary Lagrangian spheres in non-Einstein Kaehler surfaces, specifically in product surfaces of two Riemannian surfaces with different constant curvatures, identifying a unique non-minimal example when both are spheres.

Contribution

It proves the existence and uniqueness of a non-minimal Hamiltonian stationary Lagrangian sphere in certain non-Einstein Kaehler product surfaces.

Findings

Existence of a unique non-minimal Hamiltonian stationary Lagrangian sphere in the product of two spheres.

Such spheres are minimal in Kaehler-Einstein surfaces but not in the specified non-Einstein cases.

The example is explicitly constructed when both surfaces are spheres.

Abstract

Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product $\Sigma_1\times\Sigma_2$ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example is defined when the surfaces $\Sigma_1$ and $ \Sigma_2$ are spheres.