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Anton Petrunin
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If you say map instead of embedding and measure area with multiplicity, then the problem become more interesting.

The following analog of Nash--Kuiper theorem was proved by Gromov in his "Partial differential relations":

Let $f\colon M\to N$ be a short map between Riemannian manifolds of the same dimenssion. Then there is arbitrarily close length-preserving map $f_\varepsilon\colon M\to N$.

The provided map is evidently a maximizer. It is not smooth and typically has creases at everywhere dense subset, but one can smooth it keeping the area nearly the same.

Anton Petrunin
  • 45k
  • 14
  • 135
  • 299