Lemma 18.26.15.D3CEEE21DA5C Let $\Delta \subset \mathbb C$ be the unit disk. There are a compact set $K\Subset \Delta $ of Hausdorff dimension zero and a negative subharmonic function $g$ on $\Delta $, smooth and harmonic off $K$, with
\[ \{ g=-\infty \} =K,\qquad \nu (g,a)=0\quad (a\in \Delta ), \qquad g|_{\partial \Delta }=0. \]
For $n\geq 2$, put $D=\Delta ^ n$, $S=K^ n$ and
\[ h(z)=\max _{1\leq j\leq n}g(z_ j),\qquad h_ L=\max \{ h,-L\} \quad (L>0). \]
The function $h$ is locally Lipschitz off $S$, and its bounded local Monge–Ampère measure $(\mathrm{dd}^{\mathrm{c}}h)^ n$ vanishes there. The measures $(\mathrm{dd}^{\mathrm{c}}h_ L)^ n$ all have the same finite mass $M>0$. Their supports are compact and converge to $S$ as $L\to \infty $, in the sense that they lie in any given neighborhood of $S$ for all large $L$.
Proof.
Use the binary Cantor set and the nonatomic probability measure $\sigma $ of example 13.28.14, placed in a small real interval in $\Delta $. Its intervals of level $k$ have length $r_ k=\exp (-4^{k+1})$ and mass $2^{-k}$. For every $s>0$,
\[ 2^ kr_ k^ s\longrightarrow 0,\qquad 2^{nk}(\sqrt n r_ k)^ s\longrightarrow 0. \]
Covering $K$ by its $2^ k$ intervals of level $k$, and $S$ by the $2^{nk}$ products of such intervals, which have diameter at most $\sqrt nr_ k$, shows that $K$ and $S$ have Hausdorff dimension zero. Define the Green potential of the disk
\[ g(z)=\int _\Delta \log \left|\frac{z-y}{1-\bar yz}\right|^2\, \mathrm d\sigma (y). \]
Then $\mathrm{dd}^{\mathrm{c}}g=\sigma $, with the normalization of $\mathrm{dd}^{\mathrm{c}}$ fixed in the conventions. The Green kernel is nonpositive, vanishes on the boundary and is smooth away from its pole. On a fixed smaller disk it is at most $\log |z-y|^2+C$. For $z\in K$, integration over the interval of level $k$ containing $z$, together with nonpositivity of the kernel elsewhere, gives
\[ g(z)\leq 2^{-k}(2\log r_ k+C)\longrightarrow -\infty . \]
Outside $K$ the potential is finite, smooth and harmonic. Its Lelong number at $a$ is $\sigma (\{ a\} )=0$. Since $K$ stays away from the boundary, $g$ is smooth near the boundary and tends uniformly to zero there.
Near a point off $S$, some coordinate $z_ j$ lies outside $K$, and the finite value $g(z_ j)$ locally exceeds all branches $g(z_ i)$ with $z_ i$ near $K$. Hence $h$ is locally the maximum of at most $n$ pluriharmonic functions, and in particular it is locally Lipschitz. For $m\leq n$ pluriharmonic functions $a_ j$, use the smooth approximations
\[ \varepsilon \log \sum _{j=1}^ m e^{a_ j/\varepsilon }, \]
which decrease to $\max _ ja_ j$ as $\varepsilon \downarrow 0$. Their complex Hessian is a positive multiple of the weighted covariance of the forms $\partial a_ j$, with probability weights proportional to $e^{a_ j/\varepsilon }$. The centered forms satisfy one weighted linear relation, so this Hessian has rank at most $m-1<n$, and its $n$th power vanishes. Bounded decreasing convergence in theorem 5.27.1 gives the same conclusion for the maximum.
The bounded truncation $h_ L$ is continuous, since near a pole of any curve potential the corresponding branch lies below the constant $-L$. The same argument, applied to the $n+1$ branches $g(z_1),\ldots ,g(z_ n),-L$, shows that $(\mathrm{dd}^{\mathrm{c}}h_ L)^ n$ is carried by
\[ Q_ L=\{ z:g(z_ j)=-L\text{ for every }j\} . \]
Indeed, off $Q_ L$ one branch lies strictly below the maximum. Discarding it, together with any branches near poles, leaves at most $n$ pluriharmonic branches locally, so the measure vanishes there. Since $Q_ L$ is compact, the Chern–Levine–Nirenberg estimate (5.27.3) gives finite mass.
For $L,L'>0$, the functions $h_ L$ and $h_{L'}$ agree near $\partial D$, since $h$ tends uniformly to zero there. Integration by parts for bounded potentials gives
\[ (\mathrm{dd}^{\mathrm{c}}h_ L)^ n-(\mathrm{dd}^{\mathrm{c}}h_{L'})^ n =\mathrm{dd}^{\mathrm{c}}\left((h_ L-h_{L'}) \sum _{j=0}^{n-1}(\mathrm{dd}^{\mathrm{c}}h_ L)^ j \wedge (\mathrm{dd}^{\mathrm{c}}h_{L'})^{n-1-j}\right). \]
The current on the right has compact support, so its integral is zero. To see that the common mass $M$ is positive, fix $L$ and choose $\varepsilon >0$ so small that
\[ U=\{ h_ L<\varepsilon (|z|^2-n-1)\} \]
is nonempty. This set is relatively compact in $D$, because $h_ L$ tends to zero at the boundary while the quadratic function is strictly negative on $\overline D$. The comparison (5.27.7) for bounded potentials gives
\[ 0<\int _ U(\varepsilon \mathrm{dd}^{\mathrm{c}}|z|^2)^ n \leq \int _ U(\mathrm{dd}^{\mathrm{c}}h_ L)^ n\leq M. \]
Finally, $g$ is bounded on compact subsets of $\overline\Delta \setminus K$. Its sublevel sets $\{ g\leq -L\} $ therefore converge to $K$ as $L\to \infty $, and $Q_ L$ converges to $S$.
$\square$
Pome
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