$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$

Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).

Let $f_n \rightharpoonup f$ in $W^{1,2}(\M,\N) $ with $Jf_n > 0$ a.e., and suppose that the volume $V(\{x \in \M \, | \, Jf_n \le r\}) \to 0$ when $n \to \infty$, for some $0<r<1$. Does $ Jf_n \rightharpoonup Jf $ in $L^1(K)$ for every $K \subset \subset \operatorname{Int}(\M)$.?

I am fine with assuming that $f_n$ are Lipschits and injective, and that $V(f_n(\M)) \to V(\N) $.

The "higher integrability property of determinants" implies that if $\M,\N$ are open Euclidean domains, then $ Jf_n \rightharpoonup Jf $ in $L^1(K)$ for any compact $K \subset \subset \M$.

Without the assumption $V(Jf_n \le r) \to 0$, the answer can be negative even when $f_n$ are diffeomorphisms:

Take $\M=\N=\mathbb{S}^2$. Let $s: \mathbb{S}^2 \to \mathbb{R}^2 \cup \{\infty\}$ be the stereographic projection, and let $g_k(x) = k x$ for $x \in R^2$ (and $g_n(\infty) = \infty$.).

Set $ f_n = s^{-1} \circ g_n \circ s$. $f_k$ are conformal, orientation preserving, smooth diffeomorphisms and thus $ \int_{\mathbb{S}^2 }Jf_n=V(\mathbb{S}^2 )$. By conformality $\int_{\mathbb{S}^2 } |Df_n|^2 =2\int_{\mathbb{S}^2 }Jf_n$ is uniformly bounded, so $f_n$ is bounded in $W^{1,2}$, and converges to a constant function. (asymptotically we squeeze bigger and bigger parts of the sphere to a small region around the pole).

So, we do not have weak convergence of $Jf_n$ to $Jf=0$. (the Jacobians converge as measures to a Dirac mass at the pole.) The question is if by adding the non-degeneracy constraint $V(Jf_n \le r) \to 0$ we recover this 'Jacobian continuity' under weak convergence.

*(In my case of application $r=\frac{1}{4}$ but I don't think it matters).