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

We formulate and solve a degenerate elliptic boundary value problem for the $SU(\mathrm{\infty })$-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 ${\mathbb{S}}^2$.