In my lecture, I have the following theorem:
Suppose $u:(B^2 \setminus \{0\}, i) \rightarrow (M,J)$ is $J$-holomorphic with $E(u)< \infty$ (energy) and such that the image of $u$ is contained in some compact subset of $M$. Then $u$ extends to a smooth $J$-holomorphic map $u:B \rightarrow M$.
Now the definition of a bubble:
For some $J_k$ holomorphic sequence, we find a reparametrized sequence which converges locally uniformly to some $J$ holomorphic
$v: \mathbb{C} \rightarrow (M,J)$
Applying this theorem to the map \begin{align*} u: \mathbb{C} \setminus \{0\} \rightarrow M, u(z) =v(\dfrac{1}{z}) \end{align*} we conclude that $v$ extends to a nonconstant $J$-holomorphic curve $v:S^2 \rightarrow M$ and this curve is called a bubble.
Theoretically, I understand this definition but I can't really imagine what this means and also why there are pictures of the form
We have an example (which I also do not understand):
$u_k: \mathbb{C} P^1 \rightarrow \mathbb{C} P^2, [z_0:z_1] \mapsto [z_0^2:k z_0z1:z_1^2]$
For $k \rightarrow \infty$, the images converge to $\{w_0 \cdot w_1=0\}$ which is not the image of a holomorphic map $\mathbb{C} P^1 \rightarrow \mathbb{C} P^2$.
Now pick the Möbiustransformation $\varphi_k([z_0:z_1])=[k z_0:z_1]$. Then get maps $v_k = u_k \circ \varphi_k$
$v_k([z_0:z_1])=[z_0^2:z_1z_0: \dfrac{z_1^2}{k^2}]$
On compact subsets of $\mathbb{C} P^1 \setminus\{(0:1]\}$, this sequence converges to the embedding $[z_0:z1] \mapsto [z_0,z_1,0]$
Now I don't really see what this example got to do with bubbles, how is it applied here.
I know my questions is kind of long so many thanks in advance for any answer!!