Zinger fixed-domain interior Gromov compactness

Finite lower limit gives a subsequence with uniformly bounded energy and a finite limiting value $E_{\mathrm{\infty }}$. By [F3], $E_{\mathrm{\infty }}\ge \mathrm{\hslash }>0$. The energy remains the same in every conformal chart of the fixed source, since in real dimension two the inverse conformal factor in the squared norm cancels the area-form factor.

Apply [F2] with the constant coefficient sequence $J_i(z,x)=J(x)$ and the chosen target metric $g$. All its hypotheses hold: the source is the one fixed closed surface, the target is compact, the coefficients converge smoothly, and step 1.1 gives the uniform energy bound. Obtain the base solution on $\mathrm{\Sigma }$ and the finite attached sphere forest, with matched nodal values, smooth local convergence, all relative rescalings and the exact energy identity. No tame symplectic form or comparison of area with energy is needed.

If the base in step 2.1 is a constant genus-zero component with fewer than three attachment points, use [F4] to identify the genus-zero source $(\mathrm{\Sigma },j)$ biholomorphically with ${\mathbb{C}\mathbb{P}}^1$. Here genus zero means the sphere homeomorphism type. In that global coordinate, choose an original concentration point as $0$, choose gradient-maximizing points $z_i\to 0$ and scales $M_i^{-1}\to 0$ as in the normalized interior rescaling construction inside [F2], and set $h_i(z)=z_i+z\mathrm{/}{M}_i$. These are GLOBAL Möbius automorphisms of the original source. The gradient-maximizing (or Hofer-selected) rescaling in [F2] gives $\mathrm{\mid }d(u_i\circ h_i)(0)\mathrm{\mid }=1$ and a constant $C$ independent of $i$ bounding $\mathrm{\mid }d(u_i\circ h_i)\mathrm{\mid }$ on a fixed disk about zero. Its local higher-regularity estimates therefore give smooth convergence on a smaller fixed disk, with derivative norm one retained at zero. Retain that subsequence and reapply [F2] to the full maps $u_i\circ h_i$, whose energies are unchanged. The fixed-disk derivative bound excludes zero from the second extraction's concentration set, so its base has derivative norm one at zero and is nonconstant. No derivative bound on the whole global sphere is asserted. Thus the original base vertex in this second extraction is stable and survives stabilization. If the original base already was stable or nonconstant, use $h_i=\mathrm{i}\mathrm{d}$. A constant genus-one base must have an attached positive-energy tree by step 1.1, hence has at least one special point; a higher-genus constant base is already stable.

Attach the forest to its specified points of the base. The connected dual graph is a tree attached to the original base vertex, so its arithmetic genus is $g(\mathrm{\Sigma })$. Total component energy is $E_{\mathrm{\infty }}>0$, hence at least one component map is nonconstant. Contract any unstable constant genus-zero vertex: a valence-one vertex is deleted with its edge, and a valence-two vertex is deleted by joining its two neighbors at their corresponding attachment points. Those values agree because the contracted map was constant. Each operation preserves connectedness, the arithmetic genus and all energy. There are finitely many vertices, so this process terminates; a valence-zero constant component cannot be the whole remaining curve because a nonconstant component with positive energy remains. A constant genus-one base has at least one attached tree, since otherwise all energy would be zero; it is therefore stable. Constant higher-genus components are already stable. Thus [F1] identifies the result as a stable map of precisely the original genus.

The base-preserving extraction in step 3.1 supplies actual global source automorphisms. The remaining component maps and their rescalings retain the smooth convergence from [F2]. Removing finitely many constant unstable vertices only concatenates adjacent vanishing necks; all their energy and image diameters tend to zero and the target node values remain matched. It removes no positive component energy. Consequently the convergence after stabilization has exactly the component, node and no-neck properties of [F1], and the sum of its component energies remains $E_{\mathrm{\infty }}$. This proves the entire original fixed-source metric-energy Statement. No varying-domain, boundary, strip-breaking, gluing or virtual-cycle assertion is deduced.