Explicit degree-two sphere bubbling on the fixed sphere

In the affine coordinate, $u_n=(n{z}^2+1)\mathrm{/}(nz)$ has no cancellation because its numerator is $1$ at $z=0$; its rational degree is therefore two. For all target values outside the finite branch-value set, the equation $n{z}^2-nyz+1=0$ has two distinct roots. On the complement of the branch values, the map is a two-sheeted orientation-preserving covering. Applying [F3] on local inverse sheets and exhausting away from the finite branch set gives ${\int }_{{\mathbb{C}\mathbb{P}}^1}{u}_n^{\ast }{\omega }_{FS}=2{\int }_{{\mathbb{C}\mathbb{P}}^1}{\omega }_{FS}=2$; the omitted branch points have measure zero.

On every compact subset of ${\mathbb{C}\mathbb{P}}^1\setminus \{0\}$, $u_n(z)=z+(nz{)}^{-1}$ converges with all derivatives to $u_0(z)=z$. In the rescaled coordinate $z=w\mathrm{/}n$, $u_n(w\mathrm{/}n)=w\mathrm{/}n+1\mathrm{/}w$ converges as a sphere-valued holomorphic map on compact subsets of ${\mathbb{C}\mathbb{P}}^1\setminus \{\mathrm{\infty }\}$ to $u_1(w)=1\mathrm{/}w$. These are degree-one maps, so each has area one by the same change-of-variables calculation as step 1.1. Their node values agree: $u_0(0)=0=u_1(\mathrm{\infty })$.

The two nonconstant components have trivial map-preserving automorphism groups, so the nodal map is stable. The total sequence energy equals its area $2$ by [F4], while the two component energies sum to their areas $1+1=2$. Since all remaining concentration or neck energy is nonnegative, equality leaves no energy for any additional bubble or neck; hence this rescaling is the full stable Gromov limit.