The annular argument of theorem 5.39.1 also applies to a continuous potential whose modulus of continuity is not a power; what matters is a Dini condition on the square root of the modulus. This gives uniform distance estimates for families whose Hölder constants are not controlled.
Keep the coordinate domain $U\subset \mathbb C^ n$ with its Euclidean form $\omega _{\mathrm E}$ and volume $\mathrm{d}\lambda $, but now let $v\in C^0(U)$ and let $\theta $ be a smooth real $(1,1)$-form with $T=\theta +\mathrm{dd}^{\mathrm{c}}v\geq 0$. On the relatively compact charts under consideration, choose a nondecreasing function $m:(0,r_0)\to [0,+\infty )$ with $m(r)\to 0$ as $r\to 0$ such that
\[ |v(x)-v(y)|\leq m(|x-y|),\qquad |x-y|<r_0. \]
Put
06D8010B670C
5.39.5
\begin{equation} D(r)=r+\int _0^ r\frac{\sqrt{m(t)}}{t}\, \mathrm{d}t. \end{equation}
Theorem 5.39.6.21429B9E1BF3 Assume that $D(r_0/2)<+\infty $ and that $T$ is a smooth Hermitian form $\eta $ on $U\setminus A$ for a proper analytic subset $A\subset U$. Then every point of $U$ has a neighborhood $V\Subset U$ with constants $C,r_1>0$ such that, for $x,y\in V$,
\[ d_\eta ^ U(x,y)\leq C D(8|x-y|), \qquad x,y\notin A,\quad |x-y|<r_1. \]
The constants depend only on the charts and an upper bound for $|\theta |$; the dependence on $v$ enters only through $D$. All sequences in $V\setminus A$ converging to a given point of $A\cap V$ have a common limit in the metric completion. If these hypotheses hold on a finite cover of a compact connected manifold by charts, the metric completion of the regular locus is compact, and the identity of the regular locus extends to a continuous surjection of the manifold onto it. If $A$ is nonempty, the metric $\eta $ on the regular locus is incomplete.
Proof.
Write $d=2n$ and $\mu =T\wedge \omega _{\mathrm E}^{n-1}$. The same cutoff calculation as in theorem 5.39.1, with a constant subtracted from $v$, gives
\[ \mu (B(x,s))\leq C\bigl (s^ d+s^{d-2}m(2s)\bigr ). \]
This uses only the bound for $\theta $; closedness of $\theta $ is not needed. If $\tau $ is the Euclidean trace of $\eta $, then $\tau \, \mathrm{d}\lambda \leq C\mu $ on the regular locus. Polar integration bounds the average length of the segments from a regular point $x$ to the points of $B(x,R)$ by
\[ C\int _{B(x,R)}\sqrt{\tau (\zeta )} |\zeta -x|^{1-d}\, \mathrm{d}\lambda (\zeta ). \]
On an annulus of outer radius $s$, the Cauchy–Schwarz inequality and the trace bound make the contribution to this integral at most
\[ Cs^{1-d}s^{d/2} \bigl (s^ d+s^{d-2}m(2s)\bigr )^{1/2} \leq C\bigl (s+\sqrt{m(2s)}\bigr ). \]
For $s_ j=2^{-j}R$, monotonicity gives
\[ \sum _{j\geq 0}\sqrt{m(2s_ j)} \leq \frac1{\log 2}\int _0^{4R}\frac{\sqrt{m(t)}}t\, \mathrm{d}t. \]
Indeed, $\sqrt{m(2s_ j)}\log 2$ is at most the integral of $\sqrt{m(t)}/t$ over $[2s_ j,4s_ j]$, and these intervals have disjoint interiors; no doubling condition on $m$ is needed. The average length is therefore at most $CD(4R)$.
For $x,y\in V\setminus A$, average the lengths of the broken segments $[x,z]\cup [z,y]$ over $z$ in the ball of radius $|x-y|/2$ about the midpoint of $x$ and $y$, applying the preceding bound with $R=2|x-y|$ at each endpoint. Since the volume of this ball is comparable to $R^ d$, some broken path has length at most $CD(8|x-y|)$. As in the proof of theorem 5.39.1, almost every such path avoids $A$, and for small $V$ it stays inside $U$.
Since $D(r_0/2)<+\infty $, we have $D(r)\to 0$ as $r\to 0$. Hence any two regular sequences converging to the same point are Cauchy and have the same limit in the completion. The argument for the completion in the proof of theorem 5.39.1 applies with this modulus. A positive lower bound for $\eta $ near a regular point prevents a sequence converging to a point of $A$ from converging in the metric to the regular point. This also proves incompleteness when $A$ is nonempty.
$\square$
Corollary 5.39.7.380EB17A4350 In the setting of theorem 5.39.6, let $V$ be as in the theorem. For $0<\beta \leq 1$, a modulus $m(r)\leq Cr^\beta $ gives $d_\eta ^ U(x,y)\leq C'|x-y|^{\beta /2}$ for sufficiently close $x,y\in V\setminus A$. If instead
\[ m(r)\leq C(-\log r)^{-a},\qquad a>2, \]
then for sufficiently close $x,y\in V\setminus A$,
\[ d_\eta ^ U(x,y)\leq C'(-\log |x-y|)^{1-a/2}. \]
These are only sufficient conditions; we do not claim that the Dini condition is necessary.
Proof.
Integrating the square root of the power modulus gives a multiple of $r^{\beta /2}$. For the logarithmic modulus, the substitution $s=-\log t$ gives
\[ \int _0^ r\frac{(-\log t)^{-a/2}}t\, \mathrm{d}t =\frac{(-\log r)^{1-a/2}}{a/2-1}. \]
In either case the term $r$ in $D(r)$ is bounded by the asserted rate for small $r$. The theorem now gives both estimates; the fixed dilation of the argument of $D$ changes only the constants.
$\square$
Corollary 5.39.8.D3B517FBCE4F Let $\omega _0$ be a Hermitian form on $X$ and $d_{\omega _0}$ the distance of the associated Riemannian metric. For each $j$, let $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}v_ j\geq 0$, where $\theta _ j$ is a smooth real $(1,1)$-form, $v_ j$ is continuous, and $T_ j$ is a smooth Hermitian form off a proper analytic subset $A_ j$. Assume
\[ |\theta _ j|_{\omega _0}\leq \Lambda ,\qquad |v_ j(x)-v_ j(y)|\leq m(d_{\omega _0}(x,y)), \]
with one fixed $\Lambda $ and one nondecreasing modulus $m$ tending to zero such that the function in (5.39.5) satisfies $D(r_0/2)<+\infty $. By theorem 5.39.6, the identity of $X\setminus A_ j$ extends to a continuous surjection of $X$ onto the metric completion of $X\setminus A_ j$; let $d_ j$ be the pseudometric on $X$ obtained by pulling back the completed distance. Then the diameters of the $d_ j$ are uniformly bounded, and a subsequence converges uniformly on $X\times X$ to a continuous pseudometric. If $T_ j\geq c\omega _0$ on the regular loci for one $c>0$, the limit is a metric and dominates $\sqrt c\, d_{\omega _0}$.
Proof.
A finite fixed chart cover and the theorem give common constants $C,r_1>0$ such that
\[ d_ j(x,y)\leq CD(Cd_{\omega _0}(x,y)), \qquad d_{\omega _0}(x,y)<r_1. \]
For a pair meeting $A_ j$, approximate it by regular pairs in the manifold topology. The completed distance and $D$ are continuous, so the same estimate holds for all such pairs in $X\times X$. No uniform bound on the geometry of the analytic sets enters the constants. Dividing a minimizing geodesic of $d_{\omega _0}$ into a uniformly bounded number of short pieces and using the triangle inequality gives a uniform diameter bound. The triangle inequality also gives
\[ |d_ j(x,y)-d_ j(x',y')| \leq d_ j(x,x')+d_ j(y,y'), \]
so the distance functions are equicontinuous. The Arzelà–Ascoli theorem gives a subsequence $d_{j_ k}$ converging uniformly on $X\times X$ to a function $d_\infty $. Nonnegativity, symmetry, vanishing on the diagonal and the triangle inequality pass to the limit. The lower bound $T_ j\geq c\omega _0$ gives $d_ j\geq \sqrt c\, d_{\omega _0}$, first for regular pairs and then for all pairs, which proves the last assertion.
Without this lower bound the limit may degenerate. For $\varepsilon _ j\downarrow 0$, take $\theta _ j=\varepsilon _ j\omega _0$ and $v_ j=0$, with $A_ j=\varnothing $. These data satisfy the hypotheses with $m(r)=r^2$, and $d_ j=\sqrt{\varepsilon _ j}\, d_{\omega _0}$ converges uniformly to zero.
$\square$
Pome
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