Boundary Value Problems for the $SU(\infty )$-Toda Equation via the Onofri Inequality

Abstract

We formulate and solve a degenerate elliptic boundary value problem for the $SU(\infty )$-Toda equation on $[0,\frac{1}{2})\times {\mathbb{S}}^2$. This constructs anti-self-dual Poincaré--Einstein metrics with circle symmetry on the $4$-ball and on complex line bundles over ${\mathbb{S}}^2$.

40 pages. Comments are welcome