25.12.3 Big backgrounds under mass boundsBCD8D94DC861

Li's theory of equations with general measures [Theorems 6.1–6.2 and Corollary 6.1, XL26] assumes that the manifold has bounded mass and that the Hermitian metric has positive lower mass. It gives theorem 10.31.3 and the constructions with a model reference in subsection 10.31.43. Under bounded mass, positivity of the lower volume of the model reference implies that the metric has positive lower mass, so expressing the hypothesis through the reference reorganizes this setting without enlarging it.

The relative foundations of this theory use the domination, stretching and obstacle estimates discussed in section 25.11. In (10.31.27), the required rooftop envelope is constructed before lemma 13.19.4 is applied to the relative projection.

Moment and continuity estimates connect these operations on envelopes with existence. The compactness argument with first moments, using (10.31.10), follows [Proposition 4.2, XL26]: one chooses a reference potential in the same singularity type whose mass is sufficiently close to the upper volume, which need not be attained. Continuity for a fixed dominating measure is [Theorem 4.2, XL26]; there the finite rooftop envelopes use only the envelope class, while the two-sided bounds for the mass prevent collapse and control the tails.

The closure theorem for varying references, theorem 10.31.30, is [Corollary 4.1, XL26]. Projection to ordered lower models gives a fixed relative class for each tail of the sequence, and an exponential contact estimate prevents collapse. The stability for positive exponents in [Corollary 6.2, XL26] assumes uniformly bounded $L^ p$ data and the ordinary type of the reference. Theorem 10.31.47 treats instead the relative full-mass class: the models converge for the volume comparison with a common positive lower bound for their lower volumes, and the densities converge in $L^1$ with respect to a fixed finite non-pluripolar measure. Extracting a summable subsequence produces the dominating measure needed for the closure theorem. For a positive exponent, uniqueness gives convergence in capacity of the whole sequence of potentials. For exponent zero, the scalars and the measures converge, every $L^1$ cluster point of the normalized potentials solves the limiting equation, and the corresponding subsequence converges in capacity. In both cases, identifying the right-hand sides in $L^1$ gives convergence of the measures in total variation.

For the minimal reference, the equations with general measures in subsection 10.31.21 are [Theorems 5.2–5.3, XL26]. The estimate for higher moments in subsection 10.31.13 follows the concave-weight argument of [Proposition 5.1 and Lemma 5.1, XL26], with the integrability against the fixed singular measure stated explicitly. The smoothing of measures used in the existence proof is collected in lemma 5.44.12, lemma 5.44.10.

The conclusion of corollary 10.31.56, that for $L^ p$ data the solutions have the ordinary type of the reference, is [Corollary 6.1, XL26]. The relative estimate proposition 10.31.54 starts from a solution in the envelope class and uses a uniform moment of order $m>2n+1$. The concave-weight method is related to [Theorem 5.1, XL26], which already assumes the ordinary type of the reference and uses the moment exponent $m>n$; that theorem is not used to deduce the ordinary type of an arbitrary solution in the envelope class.

Addition of measures requires a common lower bound during truncation. The weighted construction in subsection 10.31.37 is [Proposition 6.1, XL26]. For unrestricted addition, (10.31.50) follows [Theorem 6.3, XL26]; an additional rooftop envelope gives the common lower bound needed to pass through the cutoff.