Calibrated submanifolds, adiabatic limit, and gradient graphs

Abstract: We study the compactness question for certain special Lagrangians in semiflat SYZ fibrations (resp. associative submanifolds in Donaldson's proposal of collapsing coassociative K3 fibrations), and give some criterion for when gradient graphs emerge from the adiabatic limit. This gives a partial converse to the Donaldson-Scaduto proposal.

• The paper establishes a uniform linear mass bound for calibrated cycles under semiflat SYZ fibrations and collapsing coassociative K3 fibrations.
• Gradient graph regularity is shown for special Lagrangian cycles and conditionally for associative cycles.
• Both settings are consistent and adaptable to real-world applications via controlling volume, including perturbational and stability results.

This paper by Yang Li addresses a compactness question in calibrated geometry: under what hypotheses must balanced gradient graphs arise as adiabatic limits of special Lagrangian cycles in semiflat SYZ fibrations, and of associative cycles in Donaldson's collapsing coassociative K3 fibrations. The main theorem establishes that any sequence of such calibrated cycles satisfying a uniform linear mass bound subsequentially converges to a $\Lambda$-weighted gradient cycle, giving a partial converse to the Donaldson–Scaduto gluing proposal. The result is proved uniformly for both settings, with the special Lagrangian and associative cases treated in parallel throughout.

Background: three manifestations of gradient graphs

The paper situates its question within a well-known phenomenon: when an ambient manifold collapses to a lower-dimensional base, calibrated submanifolds often project to graphs. The simplest instance is tropical geometry, where holomorphic curves in $(\mathbb{C}^*)^2$ under rescaled logarithm maps converge to tropical curves—the non-smooth locus of a tropical polynomial. This case is thoroughly developed via Mikhalkin's correspondence theorem, Parker's exploded manifolds, and the Gross–Siebert program; the paper does not attempt to improve on it but uses it as a model.

The second setting is a semiflat SYZ fibration $X = T^n \times B$ over a contractible domain $B \subset \mathbb{R}^n$ carrying a Hessian metric $g_{ij} = \partial_i \partial_j u$, with Kähler form $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$ and holomorphic volume form $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$. Special Lagrangians of phase $-\pi/2$ whose projection is a thickening of a one-dimensional graph are modelled near edges by $T^{n-1}$-bundles following gradient flowlines of linear functions determined by their homology classes $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$, with a balancing condition $(\mathbb{C}^*)^2$0 at vertices. Such data constitute a balanced gradient graph.

The third setting follows Donaldson's program for $(\mathbb{C}^*)^2$1-manifolds with adiabatic coassociative K3 fibrations. Given a positive section $(\mathbb{C}^*)^2$2 (with the technical assumption that the spanned positive 3-plane is not orthogonal to any $(\mathbb{C}^*)^2$3-class), Torelli determines hyperkähler triples on the fibres, and a closed $(\mathbb{C}^*)^2$4-structure $(\mathbb{C}^*)^2$5 is constructed—closed but not generally coclosed, so not torsion-free. Donaldson–Scaduto proposed producing associative submanifolds from balanced gradient graphs with weights $(\mathbb{C}^*)^2$6. The present paper proves the converse direction: compactness forces such graph structures to emerge.

Main compactness theorem

The key hypothesis is the uniform linear volume bound: for each compact $(\mathbb{C}^*)^2$7, the rescaled masses satisfy $(\mathbb{C}^*)^2$8 in the special Lagrangian case, and $(\mathbb{C}^*)^2$9 in the associative case. The intuition is that the limiting object grows only along one base dimension while fibre directions have scale $X = T^n \times B$0. Notably, for globally defined examples this bound reduces to controlling a single homology-dependent integral, making it checkable in practice.

The main theorem states that under this hypothesis, after passing to a subsequence there exists a $X = T^n \times B$1-weighted gradient cycle $X = T^n \times B$2 (with $X = T^n \times B$3 or $X = T^n \times B$4 respectively) such that:

1. The pushforward measures $X = T^n \times B$5 (resp. $X = T^n \times B$6) converge weakly to the density measure $X = T^n \times B$7.
2. Pairings against fibrewise-closed forms converge to $X = T^n \times B$8.
3. On any open subset properly contained in $X = T^n \times B$9, the projections $B \subset \mathbb{R}^n$0 converge in Hausdorff distance to $B \subset \mathbb{R}^n$1.

The method is robust against small $B \subset \mathbb{R}^n$2 perturbations of the ambient structure: if the closed $B \subset \mathbb{R}^n$3-structure deviates by $B \subset \mathbb{R}^n$4, all conclusions persist. This matters because in Donaldson's torsion-free program the actual ambient structure is only approximately the model one.

Monotonicity formula

The analytic core is a monotonicity formula for the mass of calibrated cycles over macroscopic base balls. In the associative case, using the primitive $B \subset \mathbb{R}^n$5 for the product-model $B \subset \mathbb{R}^n$6 and Stokes plus coarea formulas, one obtains

$B \subset \mathbb{R}^n$7

The non-product case introduces an exponential correction factor $B \subset \mathbb{R}^n$8. The special Lagrangian version replaces $B \subset \mathbb{R}^n$9 by $g_{ij} = \partial_i \partial_j u$0, reflecting the calibration inequality $g_{ij} = \partial_i \partial_j u$1. Two consequences follow immediately: uniform upper bounds on rescaled mass at scales $g_{ij} = \partial_i \partial_j u$2, and—via standard monotonicity at scale $g_{ij} = \partial_i \partial_j u$3 on the rescaled manifolds, which have uniformly bounded geometry—uniform lower bounds whenever $g_{ij} = \partial_i \partial_j u$4 meets the support. These two-sided bounds yield a subsequential limiting measure $g_{ij} = \partial_i \partial_j u$5 whose support $g_{ij} = \partial_i \partial_j u$6 is rectifiable by Preiss's theorem, with density bounded above and below.

Construction of the limiting current

The limiting functional $g_{ij} = \partial_i \partial_j u$7 vanishes on forms with two or more base factors (gaining an extra power of $g_{ij} = \partial_i \partial_j u$8) and on fibrewise exact pieces (by Stokes), so pairings depend only on the induced lattice-valued 1-form $g_{ij} = \partial_i \partial_j u$9. This produces a closed $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$0-valued 1-current $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$1 with locally finite mass dominated by $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$2.

The integrality argument is the delicate step. At a Lebesgue point $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$3 of $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$4 with unique approximate tangent space, slicing $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$5 by the projection onto the tangent line yields classes $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$6 independent of the slice parameter; comparing against the integral representation of $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$7 shows $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$8, hence $$\omega_\epsilon = \epsilon \sum g_{ij} dx_i \wedge dy_j$$9 lies in the integral lattice. A further Stokes-based estimate then shows the mass minimizer of $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$0 is calibrated by the appropriate Kähler form $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$1, hence is an $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$2-holomorphic curve, and the tangent vector satisfies $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$3—the gradient flow condition. Consequently $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$4 is a $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$5-weighted gradient cycle and $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$6. The author notes that fixing the subsequence first, before taking local limits, avoids uncountable diagonal subsequence choices.

Regularity of gradient cycles

The second half of the paper studies $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$7-weighted gradient cycles intrinsically, drawing moral analogy with Allard–Almgren's structure theory for stationary 1-varifolds. The cycles are characterized among closed $$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$8-valued rectifiable currents by equality in integrated calibration inequalities ($$\Omega_\epsilon = \bigwedge_i (\epsilon\, dy_i - \sqrt{-1}\, dx_i)$$9 versus weighted mass in the special Lagrangian case; $-\pi/2$0 versus $-\pi/2$1 in the associative case). This yields local minimization of the weighted length functional, an almost-stationarity estimate bounding the generalized mean curvature in $-\pi/2$2, a monotonicity formula with density $-\pi/2$3 well-defined everywhere, and local noncollapsing $-\pi/2$4.

A caveat emphasized by the paper: unlike classical calibrated currents, the density $-\pi/2$5 need not be integer-valued and can vary continuously, even though $-\pi/2$6 takes values in the discrete lattice $-\pi/2$7.

The tangent cone analysis uses White's compactness theorem for flat chains valued in discrete normed groups. Every tangent cone has the form

$-\pi/2$8

is unique, and satisfies $-\pi/2$9 with each $T^{n-1}$0. Uniqueness follows from finiteness of candidate cones together with connectedness of the space of tangent cones in the flat topology.

The local structure theorem decomposes $T^{n-1}$1 near any point into finitely many pieces $T^{n-1}$2, each supported in a cone around a ray, with second-order tangency $T^{n-1}$3, and vertical slices summing to $T^{n-1}$4 with total weight exactly $T^{n-1}$5. The proof combines a cone-crossing lemma—derived from Kirchhoff-type identities for $T^{n-1}$6 and a counting estimate $T^{n-1}$7—with the monotonicity formula.

Two regularity conclusions follow:

• Special Lagrangian case: $T^{n-1}$8 is always a locally finite embedded graph with edges being gradient flowlines. Here the argument is clean because distinct classes $T^{n-1}$9 give distinct directions $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$0 unless they are positive rational multiples.
• Associative case: $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$1 is locally a finite embedded graph provided a technical assumption holds at $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$2: any two classes $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$3 admitting holomorphic representatives for a common complex structure, both of area at most $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$4, must be proportional. Under a genericity assumption on $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$5—that the support avoids the codimension-two locus where competing gradient fields become parallel—this assumption holds, and $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$6 is locally a balanced gradient graph.

Limitations and open questions

The paper is explicit about what remains unresolved. First, can the technical assumption on $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$7 classes in the associative regularity theorem be dropped? The analogy with Allard–Almgren is instructive: without discreteness of density they constructed points of infinite complexity. Here the conceivable pathology is several approximately parallel gradient flowlines with distinct weights accumulating infinitely many intersections. The author observes optimistically that gradient flowlines obey first-order equations and tangent directions are finite in number, raising the possibility that infinite complexity cannot occur, perhaps provable under real-analyticity or genericity of $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$8—but this is left open.

Second, the theorem captures only the homological information of the limit. Whether $\delta(e) \in H_{n-1}(T^n,\mathbb{Z})$9 is itself an integral current, and whether its fibrewise slices along flowlines are genuine holomorphic curves (resp. sums of translated flat subtori) rather than $(\mathbb{C}^*)^2$00-weighted averages over moduli, depends on controlling oscillation of $(\mathbb{C}^*)^2$01 in the fibre direction over macroscopic base scales. If this oscillation is too wild, fibrewise slices may fail to be integral cycles. The $(\mathbb{C}^*)^2$02-valued current $(\mathbb{C}^*)^2$03 is better behaved precisely because homology classes cannot oscillate wildly—a distinction the paper makes clearly.

One should also note the standing assumptions: the $(\mathbb{C}^*)^2$04-structure is closed but not torsion-free (torsion-freeness would require $(\mathbb{C}^*)^2$05 to be a maximal submanifold in $(\mathbb{C}^*)^2$06), and the base is assumed contractible with the analysis purely local.

Conclusion

The paper converts a heuristic expectation—that calibrated submanifolds in collapsing geometries project to balanced gradient graphs—into a rigorous compactness theorem, valid simultaneously for special Lagrangians in semiflat SYZ fibrations and associatives in Donaldson's adiabatic K3 fibrations. The combination of a calibrated monotonicity formula on the base, White-style lattice-valued current compactness, and a tangent cone analysis adapted to first-order gradient dynamics yields full graph regularity in the special Lagrangian case and conditional regularity in the associative case. The partial converse to Donaldson–Scaduto is thus established at the level of currents and measures, while the finer question of integral fibrewise structure and the removal of the genericity assumption remain open.