[proofplan] We apply the Choi-Schoen compactness package for embedded minimal surfaces with uniform area and genus bounds on a closed ambient three-manifold. The area and genus hypotheses give a finite curvature-concentration set; away from this set, local curvature estimates and graphical compactness give a smoothly convergent subsequence. The local graphical limits patch uniquely on overlaps to form a smooth embedded minimal lamination of the punctured manifold. At precisely those concentration points where a specified removable-singularity criterion applies to the limiting punctured lamination, that criterion extends the lamination smoothly across the point. [/proofplan]

[step:Extract a finite curvature-concentration set from the area and genus bounds]Let $A_0>0$ and $g_0\in\mathbb N$ be the constants in the theorem statement. For each $k\in\mathbb N$, let $|A_{\Sigma_k}|: \Sigma_k\to[0,\infty)$ denote the norm of the second fundamental form of the embedded minimal surface $\Sigma_k\subset M$ with respect to $g$. We use the finite-curvature-concentration theorem of Choi-Schoen for closed embedded minimal surfaces in a closed three-manifold with uniform area and genus bounds. This external local input is used in its restricted form: it supplies only the finite exceptional set and curvature control away from that set, not the global lamination compactness conclusion being proved here. Applied to the sequence $(\Sigma_k)_{k\in\mathbb N}$, it gives a subsequence, still denoted $(\Sigma_k)_{k\in\mathbb N}$, and a finite set $\mathcal S\subset M$ such that for every compact set $K\subset M\setminus\mathcal S$ there is a constant $C_K<\infty$ satisfying \begin{align*} \sup_{x\in K\cap\Sigma_k}|A_{\Sigma_k}|(x)\le C_K \end{align*} for all sufficiently large $k$. The hypotheses of this curvature-concentration input are verified directly: $M$ is closed, hence compact without boundary; each $\Sigma_k$ is closed, embedded, and minimal; and the uniform bounds $\mathcal H^2(\Sigma_k)\le A_0$ and $\operatorname{genus}(\Sigma_k)\le g_0$ hold by assumption. The input also gives finiteness of $\mathcal S$, with cardinal bounded by a constant depending only on $(M,g)$, $A_0$, and $g_0$.[/step]

[guided]The first task is to isolate the only places where smooth compactness can fail. For a closed embedded minimal surface $\Sigma_k\subset M$, define \begin{align*} |A_{\Sigma_k}|: \Sigma_k&\to[0,\infty) \end{align*} by taking the pointwise norm, with respect to the ambient metric $g$, of the second fundamental form of $\Sigma_k$. Smooth local convergence of embedded minimal surfaces follows once these curvature norms are locally uniformly bounded, so the obstruction is curvature concentration. We invoke the Choi-Schoen curvature-concentration estimate only as the external local input specified in the theorem statement. This avoids circularity: at this stage we are not using the desired global lamination compactness conclusion, but only the narrower assertion that area and genus bounds force all curvature concentration into finitely many points. Its hypotheses require a compact ambient three-manifold, embedded minimal surfaces without boundary, a uniform area bound, and a uniform genus bound. These are verified as follows. The theorem statement assumes that $(M^3,g)$ is closed, so $M$ is compact and has no boundary. It also assumes that every $\Sigma_k$ is a closed embedded minimal surface in $M$. Finally, the displayed inequalities in the statement give \begin{align*} \mathcal H^2(\Sigma_k)\le A_0, \qquad \operatorname{genus}(\Sigma_k)\le g_0 \end{align*} for every $k\in\mathbb N$. The conclusion of the curvature-concentration theorem is therefore available: after passing to a subsequence, there is a finite subset $\mathcal S\subset M$ such that curvature is uniformly controlled on compact sets disjoint from $\mathcal S$. Explicitly, for every compact set $K\subset M\setminus\mathcal S$ there is a finite constant $C_K$ such that \begin{align*} \sup_{x\in K\cap\Sigma_k}|A_{\Sigma_k}|(x)\le C_K \end{align*} for all sufficiently large $k$. This is where the area and genus assumptions are used: they rule out infinitely many independent curvature-concentration points.[/guided]

[step:Use local graphical compactness away from the concentration set]Fix a point $p\in M\setminus\mathcal S$. Choose a geodesic ball $B_g(p,r_p)\subset M\setminus\mathcal S$ with compact closure in $M\setminus\mathcal S$. By the curvature bound from the previous step, there is $C_p<\infty$ such that \begin{align*} \sup_{x\in \overline{B_g(p,r_p)}\cap\Sigma_k}|A_{\Sigma_k}|(x)\le C_p \end{align*} for all sufficiently large $k$. The embedded local graphical [compactness theorem](/theorems/2748) for minimal surfaces applies on $B_g(p,r_p)$. Its hypotheses are satisfied because the surfaces are embedded and minimal, the ambient geometry is smooth on the compact ball, and the second fundamental forms are uniformly bounded there. We also verify the sheet-counting hypothesis. The graphical-radius estimate following from the bound $|A_{\Sigma_k}|\le C_p$ and the bounded ambient geometry on $\overline{B_g(p,r_p)}$ gives constants $\theta_p>0$ and $c_p>0$, depending only on $(M,g)$, $r_p$, and $C_p$, such that every graphical sheet meeting $B_g(p,r_p/2)$ contains a graphical subdisk of scale $\theta_p$ with \begin{align*} \mathcal H^2(\text{that graphical subdisk})\ge c_p. \end{align*} This lower bound is the local monotonicity or graphical-area estimate for embedded minimal disks with bounded curvature. Since the global area bound gives \begin{align*} \mathcal H^2(\Sigma_k\cap B_g(p,r_p))\le \mathcal H^2(\Sigma_k)\le A_0, \end{align*} at most $\lfloor A_0/c_p\rfloor$ such sheets can meet $B_g(p,r_p/2)$. Hence, after passing to a further subsequence depending on $p$, the sets $\Sigma_k\cap B_g(p,r_p/2)$ converge smoothly, possibly with locally finite multiplicity, to a smooth embedded minimal lamination $\mathcal L_p$ of $B_g(p,r_p/2)$.[/step]

[guided]Fix $p\in M\setminus\mathcal S$. Because $M\setminus\mathcal S$ is open, choose $r_p>0$ so that the closed geodesic ball $\overline{B_g(p,r_p)}$ is compact and contained in $M\setminus\mathcal S$. The previous step gives a finite constant $C_p$ with \begin{align*} \sup_{x\in \overline{B_g(p,r_p)}\cap\Sigma_k}|A_{\Sigma_k}|(x)\le C_p \end{align*} for all sufficiently large $k$. We now apply embedded local graphical compactness for minimal surfaces on $B_g(p,r_p)$. The required hypotheses are: embeddedness, the minimal surface equation, smooth bounded ambient geometry on the compact ball, uniform second-fundamental-form bounds, and local finiteness of the number of sheets. Embeddedness and minimality are hypotheses on $\Sigma_k$. Smooth bounded ambient geometry follows because $\overline{B_g(p,r_p)}$ is compact in the smooth Riemannian manifold $(M,g)$. The curvature bound is the displayed estimate. It remains to justify local finiteness of the sheet number. The graphical-radius estimate for embedded minimal surfaces with $|A_{\Sigma_k}|\le C_p$ in a compact ambient ball gives a radius $\theta_p>0$, depending only on $(M,g)$, $r_p$, and $C_p$, on which each sheet crossing $B_g(p,r_p/2)$ is graphical. The local monotonicity or graphical-area estimate then gives a constant $c_p>0$ such that each such sheet contributes at least $c_p$ of $\mathcal H^2$-area inside $B_g(p,r_p)$. Since \begin{align*} \mathcal H^2(\Sigma_k\cap B_g(p,r_p))\le \mathcal H^2(\Sigma_k)\le A_0, \end{align*} the number of sheets crossing $B_g(p,r_p/2)$ is bounded above by $\lfloor A_0/c_p\rfloor$. This is the precise place where the global area bound is converted into local sheet-counting control. The theorem therefore gives a subsequence, depending on $p$, for which $\Sigma_k\cap B_g(p,r_p/2)$ converges smoothly on compact subsets, with locally finite multiplicity, to a smooth embedded minimal lamination $\mathcal L_p$ of $B_g(p,r_p/2)$.[/guided]

[step:Diagonalize over a countable cover of the punctured manifold]Since $M\setminus\mathcal S$ is an open subset of the compact smooth manifold $M$, it is second countable and admits a countable collection of geodesic balls \begin{align*} \mathcal U:=\{B_g(p_j,r_j):j\in\mathbb N\} \end{align*} whose closures are compact subsets of $M\setminus\mathcal S$ and whose half-balls still cover $M\setminus\mathcal S$. Apply the local compactness conclusion successively on the balls in $\mathcal U$ and take a diagonal subsequence. Relabel this diagonal subsequence as $(\Sigma_k)_{k\in\mathbb N}$. Then for every $j\in\mathbb N$, the sequence $\Sigma_k\cap B_g(p_j,r_j/2)$ converges smoothly, possibly with locally finite multiplicity, to a smooth embedded minimal lamination $\mathcal L_j$ of $B_g(p_j,r_j/2)$.[/step]

[guided]The point of the countable cover is to replace many local subsequences by one subsequence that works everywhere on $M\setminus\mathcal S$. Since $M$ is a smooth manifold, it is second countable. The open submanifold $M\setminus\mathcal S$ is therefore second countable as well, so we may choose a countable family of geodesic balls \begin{align*} \mathcal U:=\{B_g(p_j,r_j):j\in\mathbb N\} \end{align*} with compact closures in $M\setminus\mathcal S$ and with half-balls covering $M\setminus\mathcal S$. For the first ball, local graphical compactness gives a subsequence converging on $B_g(p_1,r_1/2)$. From that subsequence, apply local compactness again on $B_g(p_2,r_2/2)$, and continue inductively. Define the diagonal subsequence by taking the $m$th term from the $m$th subsequence. For each fixed $j$, all sufficiently late diagonal terms belong to the subsequence chosen at stage $j$, so convergence on $B_g(p_j,r_j/2)$ is preserved. After relabelling, for every $j\in\mathbb N$ the sequence $\Sigma_k\cap B_g(p_j,r_j/2)$ converges smoothly, possibly with locally finite multiplicity, to a smooth embedded minimal lamination $\mathcal L_j$.[/guided]

[step:Patch the local limits into a lamination on $M\setminus\mathcal S$]On an overlap $B_g(p_i,r_i/2)\cap B_g(p_j,r_j/2)$, both $\mathcal L_i$ and $\mathcal L_j$ are smooth subsequential limits of the same diagonal sequence $(\Sigma_k)_{k\in\mathbb N}$. Smooth graphical convergence gives uniqueness of the limiting graph functions on each connected component of the overlap. Therefore the local laminations agree as subsets with the same smooth leaf charts on overlaps. The multiplicity is recorded separately as convergence data for the sequence, not as part of the lamination structure itself. Define $\mathcal L_\infty$ to be the lamination whose restriction to each $B_g(p_j,r_j/2)$ is $\mathcal L_j$. The overlap agreement makes this definition independent of $j$. Each leaf is smooth, embedded, and minimal because these properties are local and hold for every $\mathcal L_j$. Thus $\mathcal L_\infty$ is a smooth embedded minimal lamination of $M\setminus\mathcal S$.[/step]

[guided]We must check that the local limits obtained from different balls describe one global object. Let $B_g(p_i,r_i/2)\cap B_g(p_j,r_j/2)$ be a nonempty overlap. On this overlap, both $\mathcal L_i$ and $\mathcal L_j$ arise as smooth graphical limits of the same diagonal sequence $(\Sigma_k)_{k\in\mathbb N}$. Smooth graphical convergence has unique limits in each coordinate chart: if the graphs converge in every $C^m$ norm on compact subsets, the limiting graph functions and all of their derivatives are determined by the sequence. Hence the two local laminations agree on every connected component of the overlap as smooth leaf charts. Multiplicity is not part of the definition of the lamination; it is the locally finite number of sheets of $\Sigma_k$ converging to a given limiting leaf, and that number is attached to the convergence statement after the lamination has been patched. This overlap agreement lets us define $\mathcal L_\infty$ by declaring \begin{align*} \mathcal L_\infty\cap B_g(p_j,r_j/2)=\mathcal L_j \end{align*} for each $j\in\mathbb N$. The definition is independent of the chosen index because the local definitions agree wherever two balls meet. Smoothness, embeddedness, and minimality are local properties, and each $\mathcal L_j$ has those properties. Therefore $\mathcal L_\infty$ is a smooth embedded minimal lamination of $M\setminus\mathcal S$.[/guided]

[step:Verify smooth local convergence with multiplicity on compact subsets]Let $K\subset M\setminus\mathcal S$ be compact. Since the half-balls $B_g(p_j,r_j/2)$ cover $M\setminus\mathcal S$, choose finitely many indices $j_1,\dots,j_N\in\mathbb N$ such that \begin{align*} K\subset \bigcup_{\ell=1}^N B_g(p_{j_\ell},r_{j_\ell}/2). \end{align*} On each ball in this finite subcover, the diagonal subsequence converges smoothly with locally finite multiplicity to the corresponding restriction of $\mathcal L_\infty$. Smooth convergence is local, and the multiplicity is locally constant along each limiting sheet away from tangential sheet-merging because the convergence is graphical. Hence $\Sigma_k$ converges smoothly locally, possibly with multiplicity, to $\mathcal L_\infty$ on $K$. Since $K\subset M\setminus\mathcal S$ was arbitrary, the convergence holds on every compact subset of $M\setminus\mathcal S$.[/step]

[guided]Let $K\subset M\setminus\mathcal S$ be compact. The half-balls $B_g(p_j,r_j/2)$ cover $M\setminus\mathcal S$, so they cover $K$. Compactness of $K$ gives finitely many indices $j_1,\dots,j_N\in\mathbb N$ such that \begin{align*} K\subset \bigcup_{\ell=1}^N B_g(p_{j_\ell},r_{j_\ell}/2). \end{align*} On each ball in this finite subcover, the diagonal subsequence converges smoothly with locally finite multiplicity to the corresponding restriction of $\mathcal L_\infty$. Smooth convergence on $K$ is a local condition: every point of $K$ lies in one of the finitely many balls where convergence is already known. The multiplicity statement is also local. In the graphical representation, multiplicity counts the number of sheets converging to a limiting sheet, and the local graphical compactness theorem gives this number as locally finite. Therefore $\Sigma_k$ converges smoothly locally, possibly with multiplicity, to $\mathcal L_\infty$ on $K$. Since $K$ was arbitrary, the same conclusion holds on every compact subset of $M\setminus\mathcal S$.[/guided]

[step:Apply the specified removable-singularity criterion at eligible concentration points]Let $q\in\mathcal S$ be a point at which the removable-singularity criterion assumed in the theorem statement holds for the limiting punctured lamination. Choose a radius $\rho_q>0$ such that the geodesic ball $B_g(q,\rho_q)$ is contained in the coordinate neighbourhood where that criterion is formulated. Define the punctured local lamination \begin{align*} \mathcal L_q:=(\mathcal L_\infty\cap B_g(q,\rho_q))\setminus\{q\}. \end{align*} Then $\mathcal L_q$ is a smooth embedded minimal lamination of $B_g(q,\rho_q)\setminus\{q\}$, because $B_g(q,\rho_q)\setminus\{q\}\subset M\setminus\mathcal S$ after decreasing $\rho_q$ if necessary to avoid the other finitely many points of $\mathcal S$. The punctured-lamination removable-singularity theorem selected in the local model is now applied exactly with centre $q$, radius $\rho_q$, and punctured lamination $\mathcal L_q$. Its hypotheses are the local removable-singularity hypotheses explicitly assumed at $q$ in the theorem statement, together with the already verified facts that $\mathcal L_q$ is embedded and minimal in the punctured ball. The conclusion is that $\mathcal L_\infty$ extends across $q$ as a smooth embedded minimal lamination of $B_g(q,\rho_q)$. Applying this argument at every point of $\mathcal S$ where the stated removable-singularity criterion holds gives precisely the conditional extension asserted in the theorem. Combining the construction of $\mathcal S$, the diagonal subsequence, the limiting lamination, and the local convergence proves the compactness conclusion.[/step]

[guided]Fix $q\in\mathcal S$. We do not claim that every concentration point is automatically removable. The theorem only asserts removability at those points where a specified removable-singularity criterion applies to the limiting punctured lamination. Assume, therefore, that this criterion holds at $q$. Choose $\rho_q>0$ so that the geodesic ball $B_g(q,\rho_q)$ lies inside the coordinate neighbourhood used by the local criterion. Since $\mathcal S$ is finite, we may also decrease $\rho_q$ so that \begin{align*} B_g(q,\rho_q)\cap\mathcal S=\{q\}. \end{align*} Define \begin{align*} \mathcal L_q:=(\mathcal L_\infty\cap B_g(q,\rho_q))\setminus\{q\}. \end{align*} Because the punctured ball $B_g(q,\rho_q)\setminus\{q\}$ is contained in $M\setminus\mathcal S$, the lamination $\mathcal L_q$ is already known to be smooth, embedded, and minimal there. Now apply the removable-singularity criterion with centre $q$, radius $\rho_q$, and punctured lamination $\mathcal L_q$. The criterion's nonformal hypotheses are not hidden in the proof: they are exactly the local hypotheses assumed in the theorem statement, such as the curvature, stability, properness, or one-sided hypotheses required by the chosen local model. Together with embeddedness and minimality of $\mathcal L_q$, these are the hypotheses needed for the criterion. Its conclusion gives a smooth embedded minimal lamination of $B_g(q,\rho_q)$ whose restriction to the punctured ball is $\mathcal L_q$. Repeating this pointwise argument for every eligible point of the finite set $\mathcal S$ gives the asserted conditional extension across removable singularities. The compactness part of the theorem has already been proved on $M\setminus\mathcal S$, so this completes the proof.[/guided]