Infinite-Dimensional Flats in the Space of Positive Metrics on an Ample Line Bundle

We show that any continuous positive metric on an ample line bundle L lies at the apex of many infinite-dimensional Mabuchi-flat cones. More precisely, given any bounded graded filtration F of the section ring of L, the set of bounded decreasing convex functions on the support of the Duistermaat–Heckman measure of F embeds Lᵖ-isometrically into the space of bounded positive metrics on L with respect to Darvas’ dₚ distance for all p∈ [1,∞ ), and in particular with respect to the Mabuchi metric ($p=2$).