$2\mathrm{d}$ area maximizing short embeddings

Question

Think of a beach ball on an pool of water or sand.

Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a surface with or without boundary (in particular, the Euclidean plane with $h=e$, the standard Euclidean metric), and let $f:\mathcal{M}\longrightarrow \mathcal{N}$ be a short embedding. The energy is simply defined as the area of the embedded surface,

$$ \begin{align} E\left[ f\right]=\text{Area}\left[ f\left(\mathcal{M}\right)\right]. \end{align} $$

Is there a local/global maximizer for $E[\cdot]$ under the constraint of $f$ being short ?

(A embedding is short if $f^*h \leq g$, in the sense of quadratic forms.)

(This is, in a sense, a lower dimensional analog of the mylar balloon problem.)

Answer 1

If you say map instead of embedding and measure area with multiplicity, then the problem becomes 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.

Answer 2

If think yes, because the assumption $f^*h \leq g$ gives pointwise bounds on the derivatives of $f$ : in local charts, $|\partial_i f|^2 \leq g_{ii}$.
So up to translation, you can put the image of $f$ into a big ball, with radius depending on $h,g$ and not on $f$.
So you can consider a minimizing sequence, it will converge to something, but I'm not sure you will get a minimal surface in the end.
If you look at the equation, you will have more terms than the one for minimal surfaces because of the constraint.