$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,2}(\M,\N)$ satisfy $Jf_n > 0$ a.e., and suppose that $f_n \rightharpoonup f$ in $W^{1,2}(\M,\N) $.
Let $Jf_n dx$ be the measure on $\M$ associated with the function $Jf_n$, where $dx$ is the Riemannian volume form on $\M$. Suppose that the sequence of measures $Jf_n dx$ is uniformly bounded, i.e. $\sup_{n \in \infty}\int_{\M} Jf_n dx<\infty$.
Does a subsequence of $Jf_n dx $ weak star converge to $ Jf dx + V(\mathcal{N})\sum_{i\in I} a_i \delta_{x_i}$, where $I$ is some finite set with $a_i \in \mathbb{Z} \setminus \{0\}$ and $x_i \in \mathcal{M}$?
(since $Jf_n \ge 0$ as measures, we in fact must have $a_i > 0$.)
I think that it follows from known results in geometric measure theory (currents), but unfortunately I am not fluent in that language so I am not sure. Is there any reference for that claim?
If I understood correctly, the idea should be as follows:
Since $Jf_n dx $ is a bounded sequence of non-negative measures, it converges to some measure $\mu$. Now, a general structure theorem for currents implies that $\mu$ can be decomposed as a sum of a continuous part (which should then must be $ Jf dx$ for some reason?) and a singular part of the form $V(\mathcal{N})\sum_{i\in I} a_i \delta_{x_i}$.
I think that I found such a structure theorem-- this is "Theorem 1 (Structure Theorem)" in the book "Cartesian currents in the calculus of variations", by Giaquinta, Mariano, Modica, Guiseppe, Soucek, Jiri (Volume II, pg 363).
This theorem assumes $\M$ to be an Euclidean domain, and $\N$ is the two-sphere; I am not sure whether it holds verbatim for other manifolds.
Is the reasoning above correct?
