Problem 24.6.1B93ADA1B6064

Problem 24.6.1.B93ADA1B6064 Let $\mu $ be a smooth strictly positive probability form on $X$, and let $\mathscr C$ be a cone of backgrounds in a specified finite-dimensional space $V$ as in section 12.6, with the Hessian metric $-D^2\log \lambda _\mu $. Describe the metric completion of its unit-volume hypersurface $\mathscr S=\{ \theta \in \mathscr C:\lambda _\mu (\theta )=1\} $. Which boundary points admit a current representative, and which estimates for the minimizing Gauduchon metrics distinguish boundary points at finite distance from those at infinite distance?