25.19 Closed classes and metric completeness6DFCF19530CA

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The examples theorem 10.24.1, theorem 10.24.7, theorem 10.24.16 test the completeness question raised by Di Nezza, Guedj and Lu [v1, Theorem 3.4 and Question 3.5, pp. 14–15, DNGL26]. The first uses a big class that is not nef and an ordinary solution with minimal singularities. The second has a semipositive nef and big background, a solution of full mass and finite energy with vanishing Lelong numbers, and the prescribed nonintegrable pole with smooth strictly positive coefficient; its cusp is complete, but a disjoint null curve lies at finite normal distance. The third, theorem 10.24.16, satisfies exactly the identities of [v1, Section 3.1, pp. 12–14, DNGL26] for the initial potential, the resolution, the discrepancy, the class and the prescribed volume. Its class is not nef and has a divisorial reference with minimal singularities, while the solution adds a cusp with vanishing Lelong numbers along a disjoint curve. These examples disprove the two separate completeness assertions for the elliptic equations; whether nefness and compatibility can hold simultaneously, and whether the evolving maximal flow itself is complete, remain open. The torus examples for normal lengths in section 10.25 have instead logarithmically weakened poles whose coefficients tend to zero at the divisor.

The energy argument in section 10.24 uses the classical finite-energy class of [Section 2, BEGZ10], relative to a reference with minimal singularities; finiteness of the mixed masses and of the relative energy comes from [Proposition 1.6 and Section 2.2, BEGZ10]. Uniqueness follows from the comparison principle for full mass [Corollary 2.3, BEGZ10]: the unweighted reference measure is infinite, but the two Monge–Ampère measures being compared are finite.

In subsection 10.24.6, the potential with minimal singularities is $V_\theta =0$, and exact radial identities show that the volume form is smooth and positive across both divisors. The bounded truncations have masses $1-p(T_ k)^2$ for the square and $1-p(T_ k)$ for the mixed product on the transition shell, where $p$ is the slope of the radial profile and $T_ k$ the radius at which the truncation at level $k$ begins; their contributions to the energy tend to zero. The products are computed by smooth convex approximation and the continuity along decreasing sequences proved by Bedford and Taylor [Theorem 2.1, pp. 4–7, BT82]. This direct computation of the energy agrees with the criterion by canonical approximation of [Propositions 2.10–2.11, arXiv pp. 26–27, BEGZ10]. Here vanishing Lelong numbers do not imply ordinary minimal singularities.

In subsection 10.24.13, radial barriers and the inversion of a Volterra operator construct the cusp along the curve for a fixed smooth volume form, and explicit charts on the blowup give a smooth positive residual volume and a Kähler residual at the initial time. The reference with minimal singularities is obtained from the extension across isolated points in lemma 5.24.8. The canonical truncations at level $k$ have cutoff energy $O(k\mathrm{e}^{-k/2})$, so the continuity of bounded products and the criteria for relative energy cited above apply. The remaining smooth pulled-back metric puts the exceptional curve at finite distance. This is an auxiliary solution of an elliptic equation; it is not identified with the evolving smoothing metric.

Hölder regularity imposes a further restriction on the metric. If a positive residual current has a Hölder continuous potential and is smooth and strictly positive outside a proper analytic set, theorem 5.39.1 shows that every point of that set lies at finite distance in the metric completion; on a compact manifold, removing a nonempty analytic set therefore gives an incomplete metric. Continuity alone allows complete ends: for the prescribed radial densities of theorem 10.22.7, boundedness of the minimal remainder and finiteness of the normal length have different logarithmic thresholds. The resulting complete examples with bounded potentials solve equations with prescribed densities; they do not have the compatibility with the flow discussed above.

The mechanism of Dang [v2, Proposition 5.2, pp. 16–17, Dang26], which bounds distances through a Dini condition on the square root of the modulus of continuity of the potential, is the subject of theorem 5.39.6. The trace calculation given here keeps the smooth background through the additive term in (5.39.5) and needs no doubling condition. It gives logarithmic rates and uniform limits of pseudometrics along subsequences in corollary 5.39.7, corollary 5.39.8; a common positive lower bound for the metrics is needed to exclude collapse.