$\newcommand{\M}{\mathcal{M}} \newcommand{\N}{\mathcal{N}} \newcommand{\VolM}{\text{Vol}_{\M}} \newcommand{\VolN}{\text{Vol}_{\N}}$ This question is mainly a reference request.

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 easily seen to be

$$ \delta\big((df)^{-T}\big)=0,$$

where $\delta:\Omega^1(M,f^*TN) \to \Gamma(f^*TN)$ is the adjoint of the exterior derivative $$d_{\nabla^{f^*T\N}}:\Gamma(f^*TN) \to \Omega^1(M,f^*TN) $$ induced by the pullback connection of the Levi-civita connection on $\N$.

(This essentially follows from a pointwise calculation; See e.g. here and here)

It is trivial to see that volume-preserving maps, i.e. maps $f$ with $\det df=1$ are symmetries of $E$, hence critical points. Similarly, every map with constant determinant is critical.

Furthermore, affine maps (in the sense $\nabla df=0$) are also critical points.

**Question:** 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?