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I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A^*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If $$ \frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right) $$ then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is $$ h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\ \frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R, \end{cases} $$ where $\sigma$ is defined by $$ \frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right). $$ See Theorem 1.8 in T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).

It seems you get the same limit if you restrict the minimization problem to diffeomorphisms with given boundary data.

I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A_*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If $$ \frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right) $$ then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is $$ h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\ \frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R, \end{cases} $$ where $\sigma$ is defined by $$ \frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right). $$ See Theorem 1.8 in T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).

It seems you get the same limit if you restrict the minimization problem to diffeomorphisms with given boundary data.

Tadeusz Iwaniec (not Henryk Iwaniec) wrote many papers regarding minimization of the n-energy so you should check his recent publications to see if you find there relevant results.

I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A^*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If $$ \frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right) $$ then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is $$ h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\ \frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R, \end{cases} $$ where $\sigma$ is defined by $$ \frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right). $$ See Theorem 1.8 in T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).

It seems you get the same limit if you restrict the minimization problem to diffeomorphisms with given boundary data.

I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A^*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If $$ \frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right) $$ then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is $$ h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\ \frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R, \end{cases} $$ where $\sigma$ is defined by $$ \frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right). $$ See Theorem 1.8 in T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).

It seems you get the same limit if you restrict the minimization problem to diffeomorphisms with given boundary data.

I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A^*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If $$ \frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right) $$ then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is $$ h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\ \frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R, \end{cases} $$ where $\sigma$ is defined by $$ \frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right). $$ See Theorem 1.8 in T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).

I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A^*=A(r_*,R_*)$, $0<r<R<\infty$, $0<r_*<R_*<\infty$ be annuli in the plane. If $$ \frac{R_*}{r_*}<\frac{1}{2}\left(\frac{R}{r}+\frac{r}{R}\right) $$ then the infimum of $2$-energy among all homeomorphisms is not attained. The limit of energy minimizing homeomorphisms is $$ h^o(z)=\begin{cases} r_*\frac{z}{|z|} & r< |z|\leq\sigma\\ \frac{r_*}{2}\left(\frac{z}{2}+\frac{\sigma}{\bar{z}}\right) & \sigma\leq |z|<R, \end{cases} $$ where $\sigma$ is defined by $$ \frac{R_*}{r_*}=\frac{1}{2}\left(\frac{R}{\sigma}+\frac{\sigma}{R}\right). $$ See Theorem 1.8 in T. Iwaniec, J. Onninen, n-harmonic mappings between annuli: the art of integrating free Lagrangians. Mem. Amer. Math. Soc. 218 (2012), no. 1023 (available on arXiv).

It seems you get the same limit if you restrict the minimization problem to diffeomorphisms with given boundary data.