$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Vol}{\operatorname{Vol}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\Volm}{\operatorname{Vol}_{\M}}$ $\newcommand{\Voln}{\operatorname{Vol}_{\N}}$

Let $\M,\N$ be smooth, connected, oriented, compact $n$-dimensional Riemannian manifolds. Let $u_k,u \in W^{1,n}(\M,\N)$ be Lipschitz and satisfy $u_k \to u$ in $W^{1,n}(\M,\N)$. (strong convergence).

Is it true that $Ju_n \to Ju$ strongly in $L^1(\M)$?

I can prove that $|Ju_n| \to |Ju|$ strongly in $L^1(\M)$ (see below), so if we can prove that $Ju_n \to Ju$ a.e. we are done.

I tried to prove that $Ju_n \to Ju$ a.e. by using local coordinates, but this doesn't seem trivial; $u_k$ does not necessarily converges uniformly to $u$, so it is not clear how to do that. (Note that the values of $Ju_k,Ju$ at a point $p$ depend upon the images $u_k(p),u(p)$, unlike in the Euclidean case).

I use the definition $W^{1,n}(\M,\N)=\{ u \in W^{1,n}(\M,\R^D) , u(x) \in \N a.e.\}$, where $\N$ is implicitly assumed to be isometrically embedded in $\R^D$ via some embedding $i$. $W^{1,n}(\M,\N)$ inherits the notion of strong convergence from the ambient space $W^{1,n}(\M,\R^D)$.

The Jacobians are defined via the Riemannian and orientation structures, i.e. by requiring $u_k^*\Voln=(Ju_k) \Volm$ where $\Volm,\Voln$ are the Riemannian volume forms of $\M$ and $\N$ respectively.

Proof that $|Ju_n| \to |Ju|$ strongly in $L^1$:

$u_k \to u$ in $W^{1,n}(\M,\N)$ means $i \circ u_k \to i \circ u$ in $W^{1,n}(\M,\R^D)$, so in particular $d(i \circ u_k) \to d(i \circ u)$ in $L^{n}$. (we regard $d(i \circ u_k)$ as maps $T\M \to T\R^D$.)

A vector bundle map $L:T\M \to T\R^D$ have an associated notion of "absolute value Jacobian" defined by $\Det L=\sqrt{\det(L^TL)}=\det(\sqrt{L^TL})$. (we do not have a signed Jacobian since the dimension of the target fiber space is greater than that of the source.)

Specifying this to the maps $d(i \circ u_k):T\M \to T\R^D$, we easily obtain $\Det d(i \circ u_k) \to \Det d(i \circ u)$. Finally we note that $\Det d(i \circ u_k)=|Ju_k|$.

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\Vol}{\operatorname{Vol}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\Volm}{\operatorname{Vol}_{\M}}$ $\newcommand{\Voln}{\operatorname{Vol}_{\N}}$

Let $\M,\N$ be smooth, connected, oriented, compact $n$-dimensional Riemannian manifolds. Let $u_k,u \in W^{1,n}(\M,\N)$ be Lipschitz and satisfy $u_k \to u$ in $W^{1,n}(\M,\N)$. (strong convergence).

Is it true that $Ju_k \to Ju$ strongly in $L^1(\M)$?

I can prove that $|Ju_k| \to |Ju|$ strongly in $L^1(\M)$ (see below), so if we can prove that $Ju_k \to Ju$ a.e. we are done.

I tried to prove that $Ju_n \to Ju$ a.e. by using local coordinates, but this doesn't seem trivial; $u_k$ does not necessarily converge uniformly to $u$, so it is not clear how to do that. (Note that the values of $Ju_k,Ju$ at a point $p$ depend upon the images $u_k(p),u(p)$, unlike in the Euclidean case).

I use the definition $W^{1,n}(\M,\N)=\{ u \in W^{1,n}(\M,\R^D) , u(x) \in \N a.e.\}$, where $\N$ is implicitly assumed to be isometrically embedded in $\R^D$ via some embedding $i$. $W^{1,n}(\M,\N)$ inherits the notion of strong convergence from the ambient space $W^{1,n}(\M,\R^D)$.

The Jacobians are defined via the Riemannian and orientation structures, i.e. by requiring $u_k^*\Voln=(Ju_k) \Volm$ where $\Volm,\Voln$ are the Riemannian volume forms of $\M$ and $\N$ respectively.

Proof that $|Ju_k| \to |Ju|$ strongly in $L^1$:

$u_k \to u$ in $W^{1,n}(\M,\N)$ means $i \circ u_k \to i \circ u$ in $W^{1,n}(\M,\R^D)$, so in particular $d(i \circ u_k) \to d(i \circ u)$ in $L^{n}$. (we regard $d(i \circ u_k)$ as maps $T\M \to T\R^D$.)

A vector bundle map $L:T\M \to T\R^D$ have an associated notion of "absolute value Jacobian" defined by $\Det L=\sqrt{\det(L^TL)}=\det(\sqrt{L^TL})$. (we do not have a signed Jacobian since the dimension of the target fiber space is greater than that of the source.)

Specifying this to the maps $d(i \circ u_k):T\M \to T\R^D$, we easily obtain $\Det d(i \circ u_k) \to \Det d(i \circ u)$. Finally we note that $\Det d(i \circ u_k)=|Ju_k|$.

Edit:

Let me explain why I don't think that $Ju_n \to Ju$ a.e. is obvious: By definition, we have $$ (\Voln)_{u_k(p)}\big( (du_k)_{p}(v_1),\dots,(du_k)_{p}(v_1) \big)=(u_k^*\Voln)_p(v_1,\dots,v_n)=(Ju_k)_p (\Volm)_p(v_1,\dots,v_i), \tag{1} $$ where $v_i \in T_p\M$.

So, we need to show that $$(\Voln)_{u_k(p)}\big( (du_k)_{p}(v_1),\dots,(du_k)_{p}(v_1) \big) \to (\Voln)_{u(p)}\big( (du)_{p}(v_1),\dots,(du)_{p}(v_1) \big) \, \, \, \text{a.e,} \tag{2}$$
and we may assume that $u_k \to u$ and $d(i \circ u_k) \to d(i \circ u)$ a.e. on $\M$. Thus $d(i \circ u_k)_p(v_i) \to d(i \circ u)_p(v_i)$. The question is why does that imply the convergence $(du_k)_{p}(v_i)\to du_{p}(v_i)$ in $T\N$, which is what I think that we need in order establish the limit $(2)$.