Small components of subharmonic level sets

Definitions and mathematical statement

Let $S=\{x+iy:|x|<1,\ |y|<1\}$ be the open square. A real-valued function on $S$ is subharmonic if it is upper semicontinuous and its value at the center of every closed disk contained in $S$ is at most its average over the boundary circle. Consider subharmonic functions $u_k$ satisfying uniform convergence on the entire square: \[ \sup_{x+iy\in S}|u_k(x+iy)-x|\longrightarrow0. \] This is stronger than uniform convergence on compact subsets.

Put $D_k=\{z\in S:u_k(z)<0\}$. For sufficiently large $k$, the point $-1/2$ belongs to this open set; let $D_k^*$ be its connected component. Is it possible that, after discarding finitely many terms, for every $k$ and every $t\in(-1,1)$, \[ (D_k\setminus D_k^*)\cap \{x+it:-1<x<1\}\ne\varnothing? \] In words, the negative components other than the component containing $-1/2$ must collectively meet every horizontal cross-section of the square.

The intersecting component and the intersection point may depend on the height $t$. There is no requirement that one single small component cross every height. Uniform convergence forces the parts outside the large negative component to lie increasingly close to the vertical zero line of the limit function; the question asks whether subharmonicity nevertheless permits them at every height simultaneously.

Short English statement

Can extra negative components of subharmonic approximations to the real coordinate collectively reach every horizontal level?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700021 (AMR-036-0021), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Potential theory, Problem 2. Original source indexed by AMR-036-0021. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Problems of potential theory arising in value distribution, Problem 2. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/potvd.pdf.