Local domain slices and a countable compact exhaustion for simple closed curves

Apply [F1] to $\bar{\mathrm{\partial }}$ on the tangent complex line. Its real index is $6-6g$. Its kernel is the space of holomorphic vector fields. A nonzero such field has isolated zeros of positive integer index: locally its coefficient is $(z-p{)}^{r}h(z)$ with $h(p)\mathrm{\ne }0$, so the normalized vector winds $r$ times. Compactness and [F3] give total zero order $2-2g$. Thus for $g\ge 2$ the kernel is zero. For $g=1$, evaluation at one point is injective because any nonzero field has no zeros; the nowhere-zero differential from [F3] has a dual holomorphic field, so the complex kernel dimension is1. For $g=0$, evaluation at any three distinct points is injective, since three zeros would exceed total order2. Thus the real kernel dimension is at most6; the index6 and nonnegative cokernel dimension make it at least6. The index now gives the stated pairs $(a,b)$, constant within each genus.

Fix $k\ge 4$. The oriented $H^k$ complex structures form a Hilbert manifold: in a pointwise Cayley chart their tangent sections anticommute with $j_0$, and multiplication and inverse of matrices are smooth on the $H^k$ algebra by the finite Leibniz formula and Sobolev embedding. Choose the $H^{k+1}$ orthogonal complement $C$ of the smooth holomorphic-field kernel and choose smooth representatives of the $b$-dimensional cokernel as tangent complex-structure variations. Their pointwise Cayley chart gives a FINITE smooth family $j_t$. In an exponential chart of diffeomorphisms near the identity consider $G(v,t)=(\exp v{)}^{\ast }{j}_t$, $v\in C$. Its derivative at zero is $(v,\dot{t})\mapsto {\mathcal{L}}_v{j}_0+{\dot{j}}_t$. Under the standard identification of anticommuting endomorphisms with $(0,1)$-forms valued in $T\mathrm{\Sigma }$, the first term is $2j_0\bar{\mathrm{\partial }}v$; its range complement is exactly the chosen cokernel. Hence this derivative is a bounded isomorphism by [F1]. The map $G:H^{k+1}\times {\mathbb{R}}^b\to H^k$ is genuinely smooth: in coordinates it is $(df{)}^{-1}{j}_t(f)df$, and $j_t$ is a finite smooth family, so every derivative in $f$ only differentiates a supplied smooth coefficient and the single factor $df$. This does not assert differentiability of composition with an arbitrary variable $H^k$ tensor. By [F2], $G$ is a local diffeomorphism. The exponential chart consists of diffeomorphisms after shrinking in $C^1$: local inverses exist on a finite coordinate cover, and closeness to the identity prevents distant points from sharing an image.

We also need actual conformal coordinates varying smoothly in this finite family. In a $j_0$ coordinate, write its small smooth Beltrami coefficient as ${\mu }_t$, multiplied by a cutoff on a larger fixed disk. With $T$ from [F4], put $S=\mathrm{\partial }T$ and solve $h={\mu }_t(1+Sh)$ in $L^p$, $p>2$, by the Neumann series for $1-{\mu }_{t}S$. Shrink the parameter ball until $\mathrm{\parallel }{\mu }_t{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }S{\mathrm{\parallel }}_{L^p}<1\mathrm{/}2$. Then $w_t=z+Th$ solves $\bar{\mathrm{\partial }}{w}_t={\mu }_t\mathrm{\partial }{w}_t$. Differentiating the equation for $h$ solves each highest derivative using the SAME inverse $1-{\mu }_{t}S$; all other terms contain lower derivatives of $h$ and supplied smooth derivatives of ${\mu }_t$. Induction gives all local Sobolev orders and smooth parameter dependence, uniformly on compact parameter pieces. Small $C^1$ norm of ${\mu }_t$ makes $w_t$ $C^1$ close to $z$ by the first differentiated equation and Sobolev embedding. Thus on a smaller disk it is a diffeomorphism, with a smooth parameter-dependent inverse. Finite such charts cover the source. Transition maps are holomorphic because both coordinates solve the same Beltrami equation, as the ordinary inverse-chain-rule calculation shows. The same construction applies to any smooth structure sufficiently close to $j_0$, not only the chosen slice family. If a smooth input $j$ has an $H^{k+1}$ inverse normalizer $f$ from step 2.1, the relation $f^{\ast }{j}_t=j$ makes $f$ holomorphic in these smooth local coordinates: it is already $C^1$, and satisfies the ordinary Cauchy–Riemann equation. The elementary holomorphic/Weyl regularity in [F4] makes it smooth. This proves both smooth coordinate and smooth-normalizer assertions before any orbit-tangent argument; no degeneration/plumbing theory is imported.

We now prove the quotient assertion ONLY on smooth structures and smooth diffeomorphisms. Step 3.1 first promotes the inverse normalizer to smooth whenever its input structure is smooth. The dimension $a$ is constant by step 1.1, and at these smooth points the restricted vector-field directions remain transverse to the holomorphic kernel. Their derivative image is therefore the FULL smooth-structure orbit tangent $\mathrm{im}(v\mapsto {\mathcal{L}}_{v}j)$, of codimension $b$. No bounded $H^{k+1}\to H^k$ Lie derivative for an arbitrary rough $H^k$ structure is asserted. The $t$ coordinate of $G^{-1}$ annihilates those orbit tangents. Along any small SMOOTH isotopy pulling back a smooth structure, the path is differentiable in $H^k$, its inverse normalizers are smooth, and that derivative is zero; hence $t$ is constant. Thus nearby slice representatives related by a diffeomorphism near the identity have the same $t$. The kernel fields integrate to complete local holomorphic flows on the compact surface. Their finite compositions give all automorphisms near the identity: the map from their flow parameters and the complement $C$ to all nearby diffeomorphisms has invertible derivative, and step 2.1's uniqueness forces the complement coordinate to vanish for an automorphism. Kernel bases vary smoothly with $t$, since the constant-rank Fredholm equation solves the complement variable as a smooth graph.

The Hilbert manifold of $H^k$ complex structures is second countable: finite bundle charts reduce it to a subspace of a finite sum of separable Sobolev spaces, whose countable dense subsets are obtained by smooth cutoff approximations and rational Fourier polynomials. The neighborhoods from step 2.1 therefore have a countable subcover. Their finite smooth families represent every smooth complex structure up to a diffeomorphism. A smooth input structure yields a smooth diffeomorphism by elliptic bootstrapping of its local Beltrami equation, so no nonsmooth equivalence has been introduced. Exhaust each finite parameter ball by countably many compact sub-balls with closure inside it, and take any countable compact exhaustion of the second-countable target.

Let $V_t$ be the real $a$-dimensional holomorphic-field space. In an open set of injective points of $u_0$, choose $a\mathrm{/}2$ distinct points so that evaluation $V_0\to ⨁T_{p_i}\mathrm{\Sigma }$ is an isomorphism. This is possible by successive evaluation: a nonzero holomorphic field cannot vanish on an open set, and each new point reduces the complex kernel by exactly one until it is zero. The injective-point set is open: near one such point $u_0$ is an embedding by the constant-rank theorem, and compactness separates its image from the image of the complement of a slightly larger neighborhood. At each chosen image choose a smooth target submanifold $D_i$ of real codimension2 transverse to $d{u}_0(T_{p_i}\mathrm{\Sigma })$. In the $H^k$ map chart the equations $u(p_i)\in D_i$ define a Hilbert submanifold of codimension $a$, since point evaluations are bounded and smooth cutoff sections give arbitrary prescribed values. The holomorphic automorphism flows from step 4.1 have derivative $d{u}_0(V_0)$ transverse to these equations. A finite-dimensional implicit-function theorem therefore gives a unique nearby automorphism putting each nearby map into this slice. These normalizing points are auxiliary, not retained domain markings.

In these actual source and map slices the section ${\bar{\mathrm{\partial }}}_{j_t,J_{\lambda }}u$ is smooth $H^k\to H^{k-1}$. Locally it is the sum of first derivatives and a supplied smooth coefficient evaluated at $u$ times a first derivative; Taylor's integral remainder, Leibniz and $H^{k-1}\subset C^1$ give the differentiability and continuity of every derivative. Finite compact charts suffice. Its vertical derivative has the Cauchy–Riemann principal symbol. Adding $b$ domain parameters and restricting by the $a$ normalization conditions gives index $2n(1-g)+2c_1(u_0^{\ast }TM)+b-a$ by [F1]. When the derivative $L$ including finite $\lambda$ parameters is onto, its kernel is finite dimensional and its restriction to the kernel orthogonal complement is a bounded isomorphism onto the full $H^{k-1}$ equation space, by [F1]. Apply the actual arbitrary-Banach implicit theorem [F2] with $X=\ker L$, $Y=(\ker L{)}^{\perp }$ in the full domain and $Z=H^{k-1}(G)$, after a local bundle trivialization. It solves the complement variable as a smooth function of the kernel variable and gives the stated finite-dimensional zero manifold. Elliptic regularity gives smooth maps on this zero set. Dependence on its finite-dimensional coordinates is smooth at every higher Sobolev level: differentiate the equation, solve on the same smooth finite-dimensional kernel plus the fixed range complement, and apply the estimates in [F1] inductively. Equivalently the higher-level implicit-function charts have the same kernel and coincide by uniqueness. Thus changes involving composition with smoothly varying source diffeomorphisms are smooth ON this solution manifold, where the required extra source derivatives are available. No false smoothness of variable composition on the ambient fixed Sobolev space is used.

We justify the quotient topology as well. Suppose $u_i\to u_0$ and $u_i^{\mathrm{\prime }}\to v_0$ smoothly, with both limits nonconstant, the source structures in smooth precompact slices, and biholomorphisms $f_i$ satisfying $u_i^{\mathrm{\prime }}\circ f_i=u_i$. Give the target source surfaces smooth compatible area-one metrics. Each $f_i$ has Dirichlet energy1. These metrics converge smoothly, so its energy in the fixed limiting metric is $1+o(1)$ by uniform comparison. Apply the local extraction in [F5] in the smoothly varying conformal charts of step 3.1. This is precisely the finite-patch extraction in the fixed-source proof: coordinate changes converge smoothly, local rescalings and epsilon thresholds are uniform, and the finite atom identities match on overlaps. It yields a smooth filled base and finitely many spheres, with the exact energy identity. The base cannot be constant, since away from its finite atoms the identity $u_i=u_i^{\mathrm{\prime }}\circ f_i$ would then make $u_0$ constant on an open dense set. A nonconstant holomorphic map between closed connected Riemann surfaces has positive integer area in an area-one target: it is open, hence surjective; its isolated critical points are finite, and away from their images it is a finite covering with a positive integer number of sheets. Change of variables on those inverse charts gives that integer as its area. Consequently the base already has energy at least1, so has exactly1 and there are no bubbles. A one-sheeted holomorphic map has no branch point (a local zero of order $r$ would give $r$ sheets nearby); it is a biholomorphism. Thus $f_i$ converge smoothly globally after a subsequence and $u_0=v_0\circ f_{\mathrm{\infty }}$. This proves the equivalence relation closed. In particular, if both limits are $u_0$ with an injective point, $f_{\mathrm{\infty }}$ preserves $u_0$ and fixes its injective open set, so is the identity by [F4]. This proves local properness and rules out extra nearby identifications beyond the near-identity ones in steps 4.1 and5.1. It also makes the quotient Hausdorff: failure to separate two distinct orbits would produce equivalent pairs in shrinking ambient neighborhoods and contradict the just-proved closed-limit assertion. The charts are second countable because the ambient finite-slice Sobolev map charts are separable and the diffeomorphism quotient map is open. Their smooth compatibility follows from step 6.1.

Consider a sequence in one of the families specified in the Statement, for $J_i\to J$ smoothly. Extract its domain parameters in their compact piece and its distinguished points in compact $\mathrm{\Sigma }$. The maps have bounded first derivatives and images in compact $K_r$. In the finite smoothly varying conformal charts of step 3.1, subdivide source disks so each image stays in one fixed target chart: the first-derivative bound and a finite cover of compact $K_r$ give a uniform subdivision. Choose target coordinates making $J$ equal to $i$ at the chart center and shrink until the principal coefficient differs by at most one half, uniformly for the smoothly converging structures. The exact local higher-bootstrap supplier in [F5] now gives uniform bounds of every order on smaller disks. All hypotheses are checked: the solutions are already smooth, their first derivatives are bounded, the target-coordinate images stay in a compact chart subset, and all coefficient derivatives have uniform bounds. Arzelà–Ascoli and a diagonal subsequence now give smooth global convergence. The image constraint and the displayed distance inequalities are closed under this convergence. The second inequality makes the limiting distinguished point an injective point in the stated sense: it gives global uniqueness of its preimage and, by division by source distance along tangent directions, injectivity of its derivative. The first keeps its image in $U$. Conversely at any injective point of a smooth immersion, its local inverse gives a positive lower bound for that ratio nearby; compact separation gives one on the complement. A sufficiently large $c$ satisfies both inequalities and the derivative bound, and the compact image lies in some $K_r$. Combined with step 4.2 this proves the countable compact exhaustion. It uses only smoothly precompact local domain families, not a global Deligne–Mumford compactification.