Let us consider, a closed Riemannian surface $(\Sigma,h)$ and a compact RIemannian manifold $(N,g,)$ with dimension greater than $3$. If we a sequence of harmonic maps $u_n:(\Sigma,h) \rightarrow (N,g)$ with bounded energy, i.e. $$E(u_n)=\int_\Sigma \vert du_n\vert^2 \, dv < C,$$ it is well know that we have an energy identity, that is to say there exists an harmonic maps $u^\infty : (\Sigma,h) \rightarrow (N,g)$ and some bubble, i.e. harmonic maps $\omega_i:\mathbb{C} \rightarrow (N,g)$ such that

$$\lim_n E(u_n)= E(u^\infty)+\sum_{i} E(\omega_i).$$ My question is: does $i\geq 1$ it really occur?\ When $\Sigma=\hat{\mathbb{C}}$, the answer is clearly yes considering $$u_n(z)=(z,nz).$$ But here, the fact that the conformal group of $\hat{\mathbb{C}}$ is not compact seems to be crucial. So is there is example of bubbling when $\Sigma$ is not $\mathbb{C}$, especially in genus bigger than $2$?

Let us consider, a closed Riemannian surface $(\Sigma,h)$ and a compact Riemannian manifold $(N,g)$ with dimension greater than $3$. If we are given a sequence of harmonic maps $u_n:(\Sigma,h) \rightarrow (N,g)$ with bounded energy, i.e. $$E(u_n)=\int_\Sigma \vert du_n\vert^2 \, dv < C,$$ it is well know that we have an energy identity, that is to say there exists an harmonic map $u^\infty : (\Sigma,h) \rightarrow (N,g)$ and some bubbles, i.e. harmonic maps $\omega_i:\mathbb{C} \rightarrow (N,g)$ such that

$$\lim_n E(u_n)= E(u^\infty)+\sum_{i} E(\omega_i).$$ My question is:

Does $i\geq 1$ really occurs?

When $\Sigma=\hat{\mathbb{C}}$, the answer is clearly yes, considering $$u_n(z)=(z,nz).$$ But here, the fact that the conformal group of $\hat{\mathbb{C}}$ is not compact seems to be crucial. So is there an example of bubbling when $\Sigma$ is not $\mathbb{C}$, especially in genus bigger than $2$?