Solutions of Abreu's Equation with Rotation Invariance

Abstract

We consider a fourth order nonlinear partial differential equation in n-dimensional space introduced by Abreu in the context of Kahler metrics on toric varieties. Rotation invariant similarity solutions, depending only on the radial coordinate in Rⁿ, are determined from the solutions of a second order ordinary differential equation (ODE), with a non-autonomous Lagrangian formulation. A local asymptotic analysis of solutions of the ODE in the neighbourhood of singular points is carried out, and the existence of a class of solutions on an interval of the positive real semi-axis is proved using a nonlinear integral equation. The integrability (or otherwise) of Abreu's equation is discussed.