Stable almost-complex maps and Gromov convergence

Let $(X,J)$ be a smooth almost-complex manifold. A nodal closed Riemann surface $C$ is a connected compact complex curve whose only singularities are ordinary nodes; its normalization is a finite disjoint union of smooth compact Riemann surfaces, and each node identifies two distinct marked points on that normalization. If the normalization has components $C_1,\dots ,C_N$ and $C$ has $E$ nodes, its arithmetic genus is $g_a(C)={\sum }_{i=1}^{N}g(C_i)+E-N+1$. A stable $J$-holomorphic map is a continuous map $u:C\to X$ whose pullback to each normalized component is smooth and $J$-holomorphic, whose values agree at both branches of every node, and whose automorphism group preserving the map and nodes is finite. Equivalently, every constant genus-zero component has at least three nodal special points and every constant genus-one component has at least one; constant components of genus at least two already have finite automorphism group.

Fix a smooth Riemannian metric $g$ on $X$. For a smooth map $v:(S,j_S)\to X$, set $E_g(v)=\frac{1}{2}{\int }_S\mathrm{\mid }dv{\mathrm{\mid }}_{g,h}^{2}d{\mathrm{v}\mathrm{o}\mathrm{l}}_h$, where $h$ is any metric in the conformal class of $j_S$; in real source dimension two this is independent of $h$. For a nodal map, $E_g(v)$ is the sum of these energies over its normalized components.

Let $(\mathrm{\Sigma },j)$ be a fixed connected compact closed Riemann surface and let $u_i:(\mathrm{\Sigma },j)\to X$ be smooth $J$-holomorphic maps. A candidate limit domain $C_{\mathrm{\infty }}$ is obtained from $(\mathrm{\Sigma },j)$ by attaching finitely many rooted trees of spheres at pairwise distinct points $z_1,\dots ,z_{\mathrm{\ell }}$; every edge is a node and $g_a(C_{\mathrm{\infty }})=g(\mathrm{\Sigma })$. Let $u_{\mathrm{\infty }}:C_{\mathrm{\infty }}\to X$ be a stable $J$-holomorphic map. The sequence Gromov-converges to $u_{\mathrm{\infty }}$ exactly when the following rescaling and energy data exist.

1. There are $h_i\in \mathrm{Aut}(\mathrm{\Sigma },j)$ such that $u_i\circ h_i$ converges in $C^{\mathrm{\infty }}$ on every compact subset of $\mathrm{\Sigma }\setminus \{z_1,\dots ,z_{\mathrm{\ell }}\}$ to the base-component map $u_{\mathrm{\infty },0}$.

2. For each root sphere attached at $z_a$, choose a coordinate disk $U_a\subset \mathrm{\Sigma }$ about $z_a$ and increasing open sets $V_{a,i}\subset \mathbb{C}$ with $V_{a,i}\subset V_{a,i+1}$ and ${\bigcup }_i{V}_{a,i}=\mathbb{C}$. There are biholomorphic embeddings ${\psi }_{a,i}:V_{a,i}\to U_a$ whose images shrink to $z_a$, and $u_i\circ h_i\circ {\psi }_{a,i}$ converges in $C^{\mathrm{\infty }}$ on compact subsets away from node points to the root-sphere map, using ${\mathbb{P}}^1=\mathbb{C}\cup \{\mathrm{\infty }\}$ with its parent node at $\mathrm{\infty }$.

3. For each child sphere attached at a node of a sphere component, choose analogous expanding coordinate domains and biholomorphic rescalings into a coordinate disk about that node in the parent sphere. Compose each child rescaling with all rescalings from its ancestors. The resulting maps converge in $C^{\mathrm{\infty }}$ on compact subsets of each child sphere away from its nodes; this rule is repeated recursively through every finite sphere tree. At each node the two limiting component maps take the same value in $X$. These nested exhaustions specify every rescaling and every component of the limit.

4. With ${\mu }_{u_i\circ h_i}=\frac{1}{2}\mathrm{\mid }d(u_i\circ h_i){\mathrm{\mid }}_{g,h}^{2}d{\mathrm{v}\mathrm{o}\mathrm{l}}_h$, the reparametrized energy measures satisfy ${\mu }_{u_i\circ h_i}\rightharpoonup {\mu }_{u_{\mathrm{\infty },0}}+{\sum }_{a=1}^{\mathrm{\ell }}({\sum }_{v\in T_a}{E}_g(u_{\mathrm{\infty },v})){\delta }_{z_a}$. In particular, $E_g(u_i)\to {\sum }_v{E}_g(u_{\mathrm{\infty },v})$ over the base and every sphere vertex, so no energy is lost on any intervening neck. Unstable constant sphere components may be contracted after convergence; this preserves the arithmetic genus, component energies and matched values.

This is a definition of fixed-source closed-domain convergence, not a compactness theorem. It includes the rescaling and no-neck content of Zinger's Definition 1.1 and Theorem 1.2; it asserts no bordered-domain or varying-source result.

Symplectic and almost-complex curve data

Each vertex below represents one normalized component with its map; an edge represents a node whose two target values agree. The drawing is schematic and is not a compactness argument.