25.14 Perron geometry on Hopf manifolds94E3F780653C

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The closed-base metric inputs in subsection 17.16.15 are Darvas' Kähler completion and CAT$(0)$ theorems [v3, Theorems 6.1 and 9.2, pp. 20–26 and 30–32, DarvasCompletion17]. Capacity endpoint attainment and preservation of the squared-energy class are [v3, Theorem 5.2 and Corollary 5.4, pp. 18–20, DarvasCompletion17]. The full-mass model characterization used in proposition 17.16.32 is [v3, Theorem 4.3, pp. 16–17, DarvasCompletion17]. These theorems are applied on the connected compact Kähler base $\mathbb P^{n-1}$. The source uses $\mathrm i\partial \bar\partial $, so potentials are divided by $2\pi $ in (17.16.21); fiber integration contributes the additional factor $\sqrt{2n}$ to distance.

The smooth-endpoint distance formula used in proposition 17.16.22, proposition 17.16.29 is [v3, equation (15), p. 8, and Theorem 7.2, p. 28, DarvasCompletion17]. The bounded Dirichlet description is [v3, Section 2.2, equations (16)–(17), p. 9, DarvasCompletion17]. Section 7, p. 27, gives a bounded-potential counterexample to using the initial-velocity formula outside its regularity hypotheses. The completed action in the book is therefore defined by metric speed.

The Hopf conclusions use the elliptic fiber action and the bounded products with smooth corrections already established in the book, followed by canonical finite-level truncation. No bounded-mass property of the Hopf manifold is imposed. Group averaging proves stationarity under all smooth variations, and radial Fourier modes give the negative raw-action directions in theorem 17.16.25. The explicit diagonal geodesics and singular radial family supply exact distance and energy calculations. Their metric is on basic potentials; its ordinary-model projection is zero throughout the finite-energy domain, so it does not solve the general ordinary-model metric problem.