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

We formulate and solve a degenerate elliptic boundary value problem for the $SU(\infty)$-Toda equation on $[0,\frac12)\times\mathbb{S}^2$. This constructs anti-self-dual Poincaré--Einstein metrics with circle symmetry on the $4$-ball and on complex line bundles over $\mathbb{S}^2$.