For the normalized exponential family $(v_\lambda ,c_\lambda )$ of lemma 15.16.5, the reference interval $[V_\theta -B,V_\theta ]$ is uniform in $0<\lambda \leq 1$. Stability therefore controls differences on all of $\Omega $, and the local Hölder estimates strengthen the convergence on compact subsets. We fix the nonzero density $0\leq f\in L^ p(\mathrm{d}V)$, $p>1$, and use the constants $B,C,m$ of that lemma.
Proposition 15.17.3.D11439C1393F For the family $(v_\lambda ,c_\lambda )$ of lemma 15.16.5, the normalized potentials are relatively compact for uniform convergence of differences on $\Omega $. As $\lambda \downarrow 0$, every cluster limit $v$ is normalized, has minimal singularities, and satisfies
\[ \mathrm{MA}_\theta (v)=cf\, \mathrm{d}V, \]
where $c>0$ is the unique constant of theorem 15.16.7. Moreover,
\[ c_\lambda \longrightarrow c, \qquad c_\lambda \mathrm{e}^{\lambda v_\lambda }f \longrightarrow cf\quad \text{in }L^ p(\mathrm{d}V). \]
Every sequence of parameters in $(0,1]$ has a subsequence for which the potentials converge in $C^\beta $ on each compact subset of $\Omega $, for all
\[ 0<\beta <\frac{2(p-1)}{np+3(p-1)}. \]
Proof.
By lemma 15.16.5, all $v_\lambda $ lie in a common reference interval, and their densities are bounded in $L^ p$ by $C\| f\| _ p$. We may therefore apply corollary 15.15.3 with $\eta =V_\theta $. Given any sequence $\lambda _ j\downarrow 0$, we pass to a subsequence with $v_{\lambda _ j}\to v$ uniformly in differences on $\Omega $ and in $L^1(X)$, and with $c_{\lambda _ j}\to c_*\in [m,C]$. The limit satisfies the same reference bounds and is normalized. Off the pole set,
\[ 0\leq 1-\mathrm{e}^{\lambda v_\lambda } \leq 1-\mathrm{e}^{\lambda (V_\theta -B)}\longrightarrow 0 \]
as $\lambda \downarrow 0$. Multiplying by $f$ and using dominated convergence, we obtain $\mathrm{e}^{\lambda v_\lambda }f\to f$ in $L^ p$ for the whole family. The compactness corollary therefore identifies the limit equation as $\mathrm{MA}_\theta (v)=c_*f\, \mathrm{d}V$, and the uniqueness of the constant gives $c_*=c$. Since every subsequential limit of $c_\lambda $ as $\lambda \downarrow 0$ equals $c$, we conclude that $c_\lambda \to c$, and the stated convergence of the densities follows.
Finally, theorem 15.17.1 gives uniform local $C^\alpha $ bounds for every $\alpha $ below its endpoint, and the uniform limit inherits these bounds by pointwise convergence. For $0<\beta <\alpha $, uniform convergence and the elementary interpolation inequality on smaller coordinate balls give
\[ [v_{\lambda _ j}-v]_{C^\beta } \leq C\| v_{\lambda _ j}-v\| _\infty ^{1-\beta /\alpha } [v_{\lambda _ j}-v]_{C^\alpha }^{\beta /\alpha } \longrightarrow 0, \]
where the Hölder bounds are taken on slightly larger compact neighborhoods. The uniformly convergent subsequence given by compactness thus converges in $C^\beta $ on every compact subset of $\Omega $ and for every exponent below the endpoint; no further extraction depending on the exponent is needed.
$\square$
Corollary 15.17.4.64893D9DABFE Suppose a group $G$ of biholomorphisms of $X$ preserves $\theta $ and the measure $f\, \mathrm{d}V$. Then every $v_\lambda $ and every cluster limit as $\lambda \downarrow 0$ is $G$-invariant. If the normalized zero-exponent equation has only one $G$-invariant solution with minimal singularities, then the whole family $v_\lambda $ converges to it as $\lambda \downarrow 0$, uniformly in differences on $\Omega $ and in each local $C^\beta $ topology of the proposition.
Proof.
For $g\in G$, pullback by $g$ preserves $\theta $-plurisubharmonicity, and $V_\theta \circ g=V_\theta $ by the definition of the envelope. Pullback also preserves the positive-exponent equation, so its uniqueness gives $u_\lambda \circ g=u_\lambda $, and the normalization preserves this equality. Pullback by a fixed smooth biholomorphism is continuous on $L^1(X)$, since its Jacobian relative to the fixed smooth volume is bounded. Hence every cluster limit is invariant almost everywhere, and therefore everywhere for the canonical representatives. This argument does not require $g$ to preserve the barrier $\rho $ or the set $\Omega $.
If the invariant solution is unique, every cluster limit is the same potential. Relative compactness then implies convergence of the whole family for uniform convergence of differences on $\Omega $: a sequence staying at a fixed positive distance from that potential would have a convergent subsequence, a contradiction. The interpolation argument above upgrades this to the asserted local $C^\beta $ convergence.
$\square$
The proposition gives compactness and identifies the equation of every cluster limit, and the symmetry criterion is one way to identify the limit itself. In general, theorem 11.44.2 shows that the whole family converges, and it describes the limit through the greatest subsolution of the zero-exponent equation that satisfies a family of integral normalizations determined by how the constant $c$ depends on the density. This holds even when the zero-exponent equation has several normalized solutions.
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