$\newcommand{\M}{\mathcal{M}} \newcommand{\N}{\mathcal{N}} \newcommand{\VolM}{\text{Vol}_{\M}} \newcommand{\VolN}{\text{Vol}_{\N}}$ This question is mainly a reference request. (It is a cross-post from MSE).
Let $\M,\N$ be $d$-dimensional oriented Riemannian manfiolds, $\M$ closed.
Consider the following functional over $C^{\infty}$ orientation-preserving maps $f:\M \to \N$:
$$ E(f)=\int_M \log \det df \,\, \VolM.$$
($ \det df $ is defined by requiring $ f^*\VolN=\det df \, \, \VolM$, where $\VolM,\VolN$ are the Riemannian volume forms).
The Euler-Lagrange's equation of this functional is
$$ \delta\big((df)^{-T}\big)=0, \tag{1}$$
where $\delta:\Omega^1(M,f^*TN) \to \Gamma(f^*TN)$ is the adjoint of the exterior derivative $$\nabla^{f^*T\N}=d_{\nabla^{f^*T\N}}:\Gamma(f^*TN) \to \Omega^1(M,f^*TN) $$ induced by the pullback connection of the Levi-civita connection of $\N$.
Clearly, volume-preserving maps are symmetries of $E$, hence critical points. Similarly, every map with constant determinant is critical. Affine maps (in the sense $\nabla df=0$) are also critical.
Question: Is equation $(1)$ for volume-preserving maps used anywhere? this Does this functional have a name? Has it been studies somewhere? (Existence of critical points, their regularity, stability etc). Is there a clear geometric interpretation for $E(f)$? Are there other "obvious" critical points besides maps of constant determinant and affine maps?
