Cartesian products of the Sierpiński carpet (and similar self-similar fractals) with itself at least twice do not attain their conformal dimension.
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2 Pith papers cite this work. Polarity classification is still indexing.
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A direct proof shows that a divergence-free horizontal vector current in the Heisenberg group decomposes as a measure on horizontal curves via the horizontal Liouville theorem.
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Cartesian products of Sierpi\'nski carpets do not attain their conformal dimension
Cartesian products of the Sierpiński carpet (and similar self-similar fractals) with itself at least twice do not attain their conformal dimension.
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Smirnov Decomposition of a Horizontal Vector Charge in the Heisenberg Group
A direct proof shows that a divergence-free horizontal vector current in the Heisenberg group decomposes as a measure on horizontal curves via the horizontal Liouville theorem.