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Let $f$ be an enitre function. Define the "filled level set of $f$ as follows: $$A_M(f):=\{z\in{\mathbb C}:\ |f(z)|\le M\}$$

Theorem 1 in Topological Properties of Level-Sets of Entire Functions asserts that the "filled level set" of an entire function is a K-set (or Arakelian set, that is, a closed set with connected and locally connected at $\infty$ complement in ${\mathbb C}_\infty$).

I am interesting in translate some results on complex variables to the harmonic framework. So my question is: Does someone knows if this theorem remains true for harmonic functions in ${\mathbb R}^N$?

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  • $\begingroup$ Maybe related to this question. $\endgroup$ Nov 13, 2015 at 15:29
  • $\begingroup$ The complement of the filled level set of an entire function $\{ z:|f(z)|>M\}$ is not necessarily connected, example $f(z)=\cos z$, $M=2$. So what are you really asking? $\endgroup$ Nov 13, 2015 at 20:12
  • $\begingroup$ Its complement in the extended complex plane ${\mathbb C}_\infty$. Sorry. $\endgroup$ Nov 14, 2015 at 10:17
  • $\begingroup$ Not related, Loïc. $\endgroup$ Nov 14, 2015 at 22:29
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    $\begingroup$ @JoséAntonioPrado-Bassas: Right: the set is required to be locally connected at $\infty$, not just connected — I missed that part. Sorry! $\endgroup$ Nov 17, 2020 at 11:19

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