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 ethe mylar balloon problem.)

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.)

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 minimizer 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)

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 ethe mylar balloon problem.)