Full varying-domain compactness of stable tame holomorphic maps

Put $g_J(v,w)=\frac{1}{2}(\omega (v,Jw)+\omega (w,Jv))$. Tameness makes it positive and J-invariant. In an oriented domain orthonormal frame (e,je), holomorphicity gives du(je)=Jdu(e), and hence $E_{g_J}(u)=\int u\ast \omega =<[\omega ],\beta >=A\mathrm{.}$ On compact X this controls any fixed target Dirichlet energy uniformly; over a compact smooth tame family the comparison constants stay uniform. If $A<0$ the moduli is empty; if $A=0$ all component maps are constant. There is then no map unless β=0, because the integral class of a constant connected map is zero. For β=0 this moduli is $X\times Mba{r}_{(g,n)}$, since map stability is then domain stability, and it is compact by the domain construction. Thus assume $A>0$.

The integral-period lemma gives $h>0$ independent of domain genus and complex structure for every nonconstant closed component. There are at most $N=\mathrm{floor}(A\mathrm{/}h)$ such components in any map in the sequence. Let $V_0$ be the number of constant components. For a vertex v of the domain dual graph set $a_v=2g_v-2+s_v$, where $s_v$ counts original marks and incident half-edges. Stability gives $a_v\ge 1$ on constant vertices, while always $a_v\ge -2$ on nonconstant vertices. The exact identity ${\sum }_v$ $a_v=2g-2+n$ therefore gives $V_0\le 2g-2+n+2N\mathrm{.}$ The total number of components and nodes is bounded (the first Betti number of the dual graph is at most g). Pass to a subsequence with a fixed original graph type and component genera. This is only finite combinatorial extraction, not an a priori analytic compactness theorem.

Add at most three nonspecial distinct points to each nonconstant component, choosing enough to stabilize it as a pointed DOMAIN; the constant components are already stable. There are at most 3N added points. Pass to a subsequence with their number ℓ fixed and order them. The compact analytic stable-domain orbifold now proved from the actual point-target algebraic stack gives a convergent subsequence of these (n+ℓ)-pointed domains. Lift locally to its finite uniformizing group, passing to a subsequence if necessary. We obtain one specified universal plumbing family with parameters $(q_i,y_i)\to 0$. The thick pieces converge smoothly and each vanishing node is an actual cylinder whose length tends to infinity. The total number of such pieces and cylinders is finite. Existing persistent nodes already match exactly.

Identify compact subsets of each limiting normalization minus its node preimages with their prelimit plumbing cores. Choose holomorphic source disks there; the pulled-back maps satisfy the actual interior equation with $J_i\to J_{\mathrm{\infty }}$ smoothly and with uniform target coefficient bounds. Fix one threshold $\epsilon >0$ smaller than the interior threshold in [F3], uniform for these autonomous compact target data. Take a countable basis of relatively compact source disks, each eventually contained in a prelimit core, and extract a subsequence on which every disk energy has a limit $e(B)$. Define $Z$ by requiring $e(B)\ge \epsilon$ for every basis disk containing the point. For any finite set of distinct points of $Z$, choose pairwise disjoint basis disks around them. Their energies sum to at most the global bound $E_0$, hence $\mathrm{\mid }Z\mathrm{\mid }\le \lfloor E_0\mathrm{/}\epsilon \rfloor$, over all normalization components together. Outside $Z$ one basis disk has limiting energy below $\epsilon$; the actual interior estimate applied on smaller disks bounds first derivatives locally uniformly. Finite target nets and a countable dense source set give a locally uniformly convergent subsequence. Around each limiting image choose a fixed target chart and a constant real-linear change taking its limiting almost-complex structure to $i$. Uniform convergence and the first-derivative bounds put the nearby images in that chart with coefficient distance at most $1\mathrm{/}2$ from $i$. The interior bootstrap in [F3] gives every higher derivative bound there. A countable patch and derivative-order diagonal gives smooth convergence on compact subsets outside $Z$ and the node preimages. This argument uses no strip boundary condition.

The limit solves the limiting equation and has energy at most $E_0$ by compact exhaustion. Removability in [F3] fills each point of $Z$ and each punctured node end. Around a point $a\in Z$ choose a disk containing no other exceptional point. Further extraction gives energy limits $M(r)={\lim }_i{E}_k(u_i;B_r(a))$ for a sequence of radii decreasing to zero. Smooth convergence on every intervening compact annulus gives $M(r)-M(r^{\mathrm{\prime }})=E_k(u;B_r\setminus B_{r^{\mathrm{\prime }}})$ at these radii. Thus $m={\lim }_{r\downarrow 0}M(r)$ exists and $M(r)=m+E_k(u;B_r)$; inclusions between such disks and continuity of the filled map's energy extend this equality and existence of $M(r)$ to every smaller radius. Every such disk contains a basis disk about $a$ of limiting energy at least $\epsilon$, so $m\ge \epsilon >0$. We have now established all the off-atom convergence, smooth filling and energy-limit hypotheses of soft rescaling, rather than using its conclusion to obtain them.

Apply that supplier after translating and dilating this disk to $B_1$. It gives a sphere, finitely many child atoms, exact mass identity, equality of attaching values, and zero connecting-annulus energy and diameter. Each child mass is at least its uniform $h>0$ and at most the parent mass minus $h$. Reapply it on disjoint child disks: its own extraction and removability establish the same hypotheses for each child, and its quantum is uniform under subsequent rescalings. Every branch has depth at most $\lfloor E_0\mathrm{/}h\rfloor$ and every vertex has finitely many children, so the resulting sphere tree is finite. Summing the finite mass identities captures the entire atom. Rescaled target coefficients have autonomous limit $J_{\mathrm{\infty }}$ because their explicit source derivatives acquire the shrinking scale factors. The same disk extraction applies on any exhaustion of a translated cylindrical region used below for node clusters.

There can also be concentration at a degenerating domain node, so the preceding step alone is insufficient. Write its smoothing neck as $[0,L_i]\times S^1$, $L_i\to \mathrm{\infty }$. Fix the local-energy threshold ε below the existing small-local-energy cylinder-collapse threshold. Choose a maximal disjoint family of angular FULL-width windows of fixed length, each of energy at least ε/2. It contains at most $2E_0\mathrm{/}\epsilon$ windows. Any point whose fixed-radius patch has energy at least ε is within a fixed distance of one of these selected windows: otherwise its enclosing fixed-length full-width window is disjoint and could have been added. Enlarging selected windows by that fixed distance therefore covers every failure of the small-local-energy criterion.

Pass to a subsequence fixing their number. Declare two selected centers equivalent if their longitudinal distance stays bounded; diagonalize the finitely many pairwise distances so every inequivalent distance tends to infinity. Merge each equivalence cluster into one bounded-width center region, whose width is uniform along that subsequence. Centers staying a bounded distance from an original neck endpoint are part of its thick-side extraction, enlarging that compact side once. For every remaining cluster, translate one center to zero. Its distances to both original endpoints and to inequivalent clusters tend to infinity. The translated maps are defined on longer and longer cylinders and have bounded total energy. The local disk extraction in steps 4.1–5.1 gives a smooth limit on R×S¹ away from finitely many atoms. Fill its two finite-energy ends after the conformal changes $z=e^{(s+it)}$ and $z^{-1}$: the autonomous puncture-removability lemma gives a sphere. Treat every finite atom by the same actual soft-rescaling tree recursion. The translated sphere may be constant; its atom trees and attachments are nevertheless retained. This procedure is finite because the number of clusters and their energy-bearing atoms is bounded.

Between the finitely many cluster regions, and between them and the fixed thick ends, every stipulated fixed-radius disk has energy below ε. The already adopted lem-small-local-energy-cylinder-neck-collapse applies to these ACTUAL cylinders: J is autonomous there, so its explicit source-derivative bound is zero. It gives, for the central part at distance R from both ends, $\mathrm{energy}\le C\epsilon e^{-cR},$ $imagediameter\le Csqrt(\epsilon )e^{-cR\mathrm{/}2},$ with constants independent of its length. The same estimates apply after deleting small bubble disks from a translated cluster by the soft-rescaling supplier's annulus proof. Taking first i→∞ at fixed R, then R→∞, shows both zero lost neck energy and equality of the adjacent filled node values. This argument addresses the entire remaining neck, not merely each fixed translated window. Every long neck has now been covered by its finitely many extracted cluster cores and these vanishing connecting regions.

For the energy identity choose finite thick-core cutoffs, finite cluster cutoffs and finite bubble-tree cutoffs. They give a finite disjoint partition up to boundary circles, which have measure zero. On cores use smooth convergence; on the bubble disks use the exact soft-rescaling mass identities; on the remaining necks use step 9.1. Sum before passing to the cutoff limits. Exhausting the finitely many ends and shrinking the finitely many atom disks gives $lim{E}_{g_{J}i}(u_i)={\sum }_v{E}_{g_J\mathrm{\infty }}(u_v)=A\mathrm{.}$ If using a fixed target metric k, its energy also has the exact identity: local smooth convergence and the same k-neck bounds give the identical partition proof. No coefficient-free area identity is being substituted for an arbitrary-metric energy identity.

The n+ℓ original domain-family sections have distinct smooth limits before map bubbling. Thus at most one of these labels lies over a first thick-side energy atom, and none lies at a degenerating node of that stabilized domain. Within a bubble rescaling record the rescaled location of the label, along the same finite sequence subsequence. If it has a finite limit away from children, put it there. If it approaches a child, continue down that finite tree. If it lies in a connecting annulus at a scale separated from both endpoints, insert the corresponding zero-energy sphere with that label, the child at a fixed finite coordinate and the parent at infinity. Explicitly rescale so the label is 0 and one child is 1; the parent is ∞. The exact zero-neck-energy identity forces this inserted component map to be constant. If several scales or child groups are separated, use their finitely many relative-distance/scale comparisons in the same way. Each inserted stable ghost separates either a label or at least two already-recorded children. There are finitely many labels and finite energy-bearing vertices, so this does not create an infinite ghost chain. The case of a label converging to an existing branch point is handled by precisely this insertion rather than declaring that a mark may coincide with a node.

Attach the component maps at their matching values. The domain degeneration and the inserted sphere TREES preserve the arithmetic genus. Forget the ℓ auxiliary labels. Contract every constant rational component with at most two remaining special flags: a leaf with one mark transfers that mark to its attachment on the adjacent component (which becomes a smooth point); an unmarked leaf is deleted; an unmarked bridge identifies its two adjacent branches. A one-component constant genus-zero remainder with at most two marks, or a one-component unmarked constant genus-one remainder, cannot occur here: if all maps are constant, the original stable (g,n) inequality excludes it; if some map is nonconstant, a positive-energy component remains elsewhere. Constant genus-one components with no attachment occur only as that excluded whole-domain case. Each contraction removes a vertex, so this finite process terminates. The maps descend because every contracted component is constant and node values match. Markings remain distinct and smooth because they were distinct special flags before the contraction and a transferred label occupies the newly smoothed attachment. The result is a stable map with all n original ordered markings, of genus g.

The limiting map has finite automorphism group. Apply the explicit transverse-slice construction to its finitely many nonconstant components: select enough immersion points, choose regular target projection fibers avoiding all special values, and take compact transverse disks with boundary radii outside the countable set of preimage radii. Their full preimages are finite, nonspecial, and stabilize those components. Constant components were stabilized in step 12.1. Any map automorphism permutes this finite additional preimage set. Its subgroup fixing that set pointwise is a subgroup of the finite automorphism group of the resulting stable pointed curve. The full map automorphism group is consequently finite as well. This uses the actual slice argument, not an unproved assertion that arbitrary smooth maps have finite point preimages. Graph convergence, after adding these auxiliary labels to the limiting nonconstant components and following them on the sequence, follows from smooth core convergence, removable endpoints and the zero-diameter neck estimates. This is exactly the topology in the current Definition.

The energy identity alone does not establish the integral homology class. Fix sufficiently deep but finite cutoffs on the limiting components. Their small deleted disks map into arbitrarily small contractible target balls about the matched node values. By 9.1 the connecting neck images of $u_i$ lie in the same balls for large i. On the remaining compact cores the actual maps converge smoothly, so the short target-geodesic interpolation is a homotopy to the corresponding limiting-core map. It stays in the same ball on each boundary collar, after choosing the ball with a margin and the core cutoff sufficiently deep.

Use the finite limiting nodal domain $\tilde{C}$ before the constant-component contractions of step 12.1. On each normalization component $S_v$, delete disjoint small open coordinate disks about its node branches, retaining a nonempty compact core $K_v$. Steps 3.1–11.1 realize disjoint orientation-preserving copies $K_{v,i}$ in $C_i$, using plumbing coordinates for original cores and the recorded rescalings for bubble and ghost cores. Cut nested bubble disks in parent-before-child order: child cores lie in the removed parent disks. The complementary pieces joining paired boundary circles are annuli, being regions between nested coordinate circles or original plumbing cylinders; a persistent node instead has two disk ends meeting at that node. These exhaust the source because all concentration points and neck clusters have been included. Define ${\pi }_i:C_i\to \tilde{C}$ by the chosen identifications on the cores. Identify each annulus with $[-1,1]\times S^1$. Write its paired target disks as $z,w$, with radii $r_{-},r_{+}$, and parametrize the first boundary by $r_{-}{e}^{i\theta }$. The other boundary parametrization is $r_{+}{h}_i(e^{i\theta })$ for an orientation-reversing circle homeomorphism $h_i$, determined by the oriented collars. For $-1\le s\le 0$ map to $z=r_{-}(-s)e^{i\theta }$, and for $0\le s\le 1$ map to $w=r_{+}s{h}_i(e^{i\theta })$. At $s=0$ both give the common node. Collar parametrizations make these formulas agree with the core boundary maps. At a persistent node use the same radial extension separately on its two disk ends. The resulting map is continuous, including at self-nodes. Every interior point of $K_v$ has precisely one preimage of local orientation degree $+1$. Triangulating each normalization with node branches as vertices shows $H_2(\tilde{C};\mathbb{Z})={⨁}_v\mathbb{Z}[S_v]$: identifying only vertices changes neither two-chains nor their cycle conditions. Testing at those core points therefore gives $({\pi }_i{)}_{\ast }[C_i]={\sum }_v[S_v]$, including every bubble component.

Let $\tilde{u}:\tilde{C}\to X$ be the matched limiting map. Fix a uniform normal-geodesic radius $\iota >0$ for the smooth metric on compact $X$. Choose the deleted branch disks so their images lie in balls of radius less than $\iota \mathrm{/}10$ about the matched node values. Deep finite cutoffs and step 9.1 put each prelimit connecting piece in the same ball for all sufficiently large $i$. On these pieces $u_i$ and $\tilde{u}\circ {\pi }_i$ thus have distance less than $\iota \mathrm{/}2$; on the finitely many cores their distance tends uniformly to zero. Consequently $H_i(x,t)={\exp }_{u_i(x)}(t{\exp }_{u_i(x)}^{-1}(\tilde{u}({\pi }_i(x))))$ defines a continuous homotopy on $C_i\times [0,1]$. At persistent nodes the branch formulas agree because both endpoint maps do. The integral singular-chain prism formula and step 15.1 give $\beta =(u_i{)}_{\ast }[C_i]={\tilde{u}}_{\ast }{\sum }_v[S_v]$ in $H_2(X;\mathbb{Z})$. Each component contracted in step 12.1 is target-constant and contributes zero two-class, so this is precisely the class of the stabilized limiting map. The argument preserves torsion; equality of real periods or area alone would not suffice.

The auxiliary choices give the same local topology. For two sets of stabilizing auxiliary points, take their union, perturbing the additional choices away from the finitely many original special points. In the common marked universal family, forgetting either set is the actual domain stabilization morphism. Local smooth core convergence and zero-diameter neck convergence lift and descend under these finite forgetting maps. Points selected away from nodes of a proposed limit can be followed as actual nearby points; if labels migrate to a bubble, 11.1 records their correct stabilized location. This proves the equality of the graph neighborhood systems, rather than assuming it from the word 'topology'.

To separate two proposed stable-map limits of one sequence, add sufficiently many auxiliary smooth labels to stabilize every component of each proposed limit, choosing them away from its nodes and original labels. Follow both finite label sets on the common prelimit domains and use the compact marked-domain space to get a common limiting marked curve. Forgetting either set gives the corresponding proposed domain limit, by the Hausdorff finite-group domain quotient and the just-described stabilization maps. On every component retained by both forgetting maps, the two limiting maps agree by uniqueness of the actual core limit in Hausdorff X. A component retained by only one forgetting map is collapsed to a node or smooth point of the other. Graph convergence in that OTHER universal family makes the entire corresponding prelimit region's image tend to that single mapped point. Therefore its map in the first limit is constant. In particular a nonconstant component, which was marked and retained, cannot disappear in the other limit. The two pulled-back maps on the common curve consequently agree everywhere, including contracted components and their attachment values. Stabilizing the common map after forgetting all auxiliary labels gives the same stable map from either side. The two limits are isomorphic. The same common-chart argument gives disjoint neighborhoods of two nonisomorphic maps: otherwise choose a sequence in the countable shrinking graph neighborhoods of both and apply this uniqueness proof. Thus the topology is Hausdorff.

A countable NETWORK is enough; we do not need to assume the auxiliary-forgetting maps are open at arbitrary bubbling points. For each of the finitely many bounded domain graphs in 2.1, use the adopted smooth-domain slice construction on each normalized component. Its countable finite-dimensional smooth families represent every smooth component complex structure. Add finite-dimensional coordinates for all distinct original/node markings and finitely many Hilbert charts for $H^k$ maps into the compact target, with k≥4. The underlying map spaces are separable: local bundle charts and rational Fourier polynomials give countable dense sets, and a countable target chart cover gives a countable Hilbert atlas. Choose the countably many product charts for the finite graph. The J-holomorphic, matched-node, stable, fixed-class subset of each is a subspace of a second-countable space. Every actual stable map occurs in one of them. Sending these tuples to their isomorphism classes is continuous in the graph topology: finite strong Sobolev convergence gives $C^1$ core convergence, and this persistently nodal domain family is continuous in its actual analytic domain charts. Thus the moduli is a continuous image of a countable union of second-countable spaces. Images of their countable bases form a countable network: whenever x lies in an open O, choose a representative and a basic neighborhood contained in the inverse image of O. A network member need not itself be open; none of this assumes otherwise. The graph neighborhood definition and 17.1 give first countability at each point by fixing one auxiliary choice and rational Hausdorff radii.

A countable network makes the space Lindelöf: for each network member contained in some member of a given open cover, select one such cover member; their countable selected family still covers. The extraction and integral-class argument in steps 1.1–16.1 prove sequential compactness. A first-countable sequentially compact space is countably compact: a countable cover with no finite subcover produces points outside its first m members, and a convergent subsequence contradicts the cover member containing its limit. Combining Lindelöfness with countable compactness proves compactness for arbitrary open covers. Together with 18.1 this is compact Hausdorff. It is metrizable as well: a compact Hausdorff space is normal, by finite disjoint-neighborhood covers. For each pair of countable network members with disjoint closures choose the continuous function separating their closures, constructed by nesting dyadic open neighborhoods using normality. These countably many functions separate points, because two distinct points have open neighborhoods with disjoint closures and hence network members contained in them. Their product map embeds the compact space in [0,1]^N, a metrizable space. This proves the desired metric conclusion without falsely assuming global second countability from one graph chart. Every extraction and bound was uniform for a convergent compact smooth tame family of J, so the parameterized union over a compact interval is compact as well.