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

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.)

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 $$f_\varepsilon\colon M\to N$$.
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 $$f$$.