25.1.19 Extremal slopes and singularity cutoffs3EBE4FDCA90E

The identity for the supremum of the difference along a weak geodesic is [Proposition 4.2.2, XiaSampleBook], in the chapter on rays; for the infimum, [Proposition 4.2.3, XiaSampleBook] refers to an analogous argument. Proposition 17.9.1 gives this argument by reversing the segment and using canonical representatives; it also identifies the essential extrema with the extrema over the set where the potentials are finite.

The assertion on bounded shifts that follows [Example 4.3.1, XiaSampleBook] is justified there by comparing long segments with truncated endpoints. Proposition 17.9.2 makes this comparison quantitative: a bound $C$ at the terminal endpoint contributes at most $Cs/T$ at time $s$ on a segment of length $T$. Passing to the regularized limits as $T\to \infty $ gives invariance under bounded perturbations and monotonicity with respect to the preorder $\preceq $ of singularities. The construction of the associated rays and the formula for their energy in terms of mixed masses are not treated here.