25.1.18 Weak geodesics4C762779903D

The definition of subgeodesics on a strip and their convexity and boundary behavior (definition 17.7.1, proposition 17.7.4) follow [Definition 4.1.1 and Proposition 4.1.2, XiaSampleBook], and lemma 17.7.6, lemma 17.7.5, theorem 17.7.8 follow the arguments of [Proposition 4.1.3, Lemma 4.1.1 and Proposition 4.2.1, XiaSampleBook]. Rooftop envelopes and the identity for envelopes of mutual singularities show that the Perron segment is not identically $-\infty $ and has the prescribed endpoints. The formula of proposition 17.7.10, which expresses the Perron segment through rooftop envelopes and a Legendre transform, follows by taking supporting affine functions of the convex function of time on each fiber where it is finite; the regularization in the space variable is retained.

That the relative energy is affine along weak geodesics, theorem 17.8.3, follows the argument of [Proposition 4.3.3, XiaSampleBook] on a compact extension of the strip. The proof uses the fiber integration formula corollary 13.6.1 and integration by parts for bounded differences, theorem 17.2.5. The vanishing of the non-pluripolar Monge–Ampère measure of a maximal plurisubharmonic function, bounded or not, is lemma 17.8.1; it is proved with the homogeneous balayages of the bounded truncations. The maximality of the complexified Perron segment, lemma 17.8.2, is proved with its envelope under rotations of the annulus, or under imaginary translations of the strip. Approximating the endpoints by their decreasing truncations then extends the identity from potentials of bounded type to potentials of finite energy.