Constructions and Rigidity of Nondegenerate $\mathbb{Z}_2$ Harmonic Functions with Quadric Branching Sets

Abstract: We study nondegenerate $\mathbb Z_2$ harmonic functions on Euclidean spaces with branching loci given by ellipsoids and planar conics. We first prove that the parameter map associated with Yan's ellipsoidal family is injective. Together with Yan's surjectivity result, this identifies the ellipsoidal models, up to Euclidean motions and overall sign, with nondegenerate harmonic quadratic polynomials of index (n-1) and positive critical value. In (\mathbb R^3), we use modified ellipsoidal coordinates to construct a nondegenerate $\mathbb Z_2$ harmonic function in a neighborhood of any planar hyperbola. We then show that no global critical $\mathbb Z_2$ harmonic function with a planar hyperbola as branch locus can have finite Almgren frequency at infinity. Using modified paraboloidal coordinates, we construct a global nondegenerate $\mathbb Z_2$ harmonic function with any prescribed planar parabola as branch locus and obtain a precise asymptotic expansion and scale-down limit. Finally, we identify the parabolic model as the infinitesimal two-valued graph potential of a family of special Lagrangian three-folds constructed by Joyce.

• The paper proves Yan’s ellipsoidal coefficient map is bijective in every dimension n ≥ 3, uniquely classifying nondegenerate Z₂ harmonic functions with prescribed admissible quadratic asymptotes.
• The paper constructs a local nondegenerate model branched along planar hyperbolas and proves that no global critical example with finite Almgren frequency can exist under the stated rigidity and compactness assumptions.
• The paper gives the first global nondegenerate parabolic model on R³, derives sharp asymptotics, and identifies it as the infinitesimal potential of Joyce’s branched special Lagrangian family.

Background and motivation

A $\mathbb{Z}_2$ harmonic function on a Riemannian manifold $M$ is a section of a flat real line bundle over $M\setminus\Sigma$ with sign monodromy around a codimension-two branching set $\Sigma$, harmonic off $\Sigma$ and square-integrable with its gradient. Such objects arise as ideal boundary points in Taubes's compactification theory for $\operatorname{PSL}(2,\mathbb{C})$ connections, as renormalized limits of Fueter sections (Habibi Esfahani–Li), and as linear models in two nonlinear problems in special holonomy: branched deformations of special Lagrangian submanifolds (Donaldson's conjecture, proved by S. He) and the reflected components of branched maximal sections in Donaldson's adiabatic description of coassociative $K3$ fibrations. Near $\Sigma$, Donaldson's expansion

$$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$

defines criticality ($A\equiv 0$) and nondegeneracy ($M$0 nowhere vanishing), and these are the properties relevant to the geometric applications.

The paper, by Yuanbo Zhou, addresses the scarcity of explicit nondegenerate examples on noncompact manifolds, specifically $M$1. It has three main contributions: a parametrization theorem for Yan's ellipsoidal family, a local construction with hyperbolic branching set together with a global nonexistence result under a frequency bound, and a global construction with parabolic branching set, including its asymptotics and its identification with the infinitesimal potential of a family of Joyce's special Lagrangian three-folds.

Rigidity of the ellipsoidal family

Yan constructed nondegenerate $M$2 harmonic functions on $M$3, $M$4, branched along codimension-two ellipsoids, asymptotic at infinity to harmonic quadratic polynomials. The coefficients of the quadratic term define a map $M$5 (where $M$6 is the positive orthant) that Yan proved surjective; injectivity was known only in dimension three, via Donaldson's monotonicity argument.

The paper proves $M$7 is injective in all dimensions, hence bijective. The argument is clean: $M$8 is homogeneous of degree $M$9, and after the substitution $M\setminus\Sigma$0 it takes the form $M\setminus\Sigma$1, where $M\setminus\Sigma$2 and $M\setminus\Sigma$3 for

$M\setminus\Sigma$4

The key lemma is that $M\setminus\Sigma$5 is homogeneous of degree $M\setminus\Sigma$6 and strictly monotone: its Hessian $M\setminus\Sigma$7 is positive definite on $M\setminus\Sigma$8, since the quadratic form $M\setminus\Sigma$9 reduces to

$\Sigma$0

Strict monotonicity on the convex set $\Sigma$1 gives injectivity of $\Sigma$2, and homogeneity then upgrades this to injectivity of $\Sigma$3. Combined with Yan's surjectivity, the corollary is a complete classification: every harmonic quadratic polynomial $\Sigma$4 on $\Sigma$5 with nondegenerate Hessian of index $\Sigma$6 and positive value at its critical point is the quadratic asymptote, uniquely up to sign, of a nondegenerate $\Sigma$7 harmonic function with ellipsoidal branching set. This closes the parametrization of the only known compact branching sets in $\Sigma$8 with quadratic growth at infinity; the paper notes that the converse rigidity question—whether ellipsoids are the only such branching sets—was recently resolved affirmatively by Yang Li, Mashayekhi, and Zhang.

Local hyperbolic construction and global obstruction

The paper next constructs a nondegenerate $\Sigma$9 harmonic function in $\Sigma$0 branched along any planar hyperbola $\Sigma$1, at least locally. The technique is a modified version of Yan's ellipsoidal coordinates, adapted so that the hyperbola becomes the coordinate locus $\Sigma$2, and the complex normal coordinate satisfies $\Sigma$3. Separation of variables reduces harmonicity to a family of second-order ODEs of the form

$\Sigma$4

where $\Sigma$5 is a linear polynomial and $\Sigma$6; the freedom in $\Sigma$7 is essential, since without it no polynomial solution exists. Choosing $\Sigma$8 so that $\Sigma$9 solves the ODE, with $\operatorname{PSL}(2,\mathbb{C})$0 a root of $\operatorname{PSL}(2,\mathbb{C})$1, and applying a reduction-of-order formula yields two independent solutions $\operatorname{PSL}(2,\mathbb{C})$2 and $\operatorname{PSL}(2,\mathbb{C})$3; the combination $\operatorname{PSL}(2,\mathbb{C})$4 with $\operatorname{PSL}(2,\mathbb{C})$5, $\operatorname{PSL}(2,\mathbb{C})$6 has leading term $\operatorname{PSL}(2,\mathbb{C})$7 with $\operatorname{PSL}(2,\mathbb{C})$8 nonvanishing, giving nondegeneracy.

The construction is genuinely local: it is defined on the region $\operatorname{PSL}(2,\mathbb{C})$9. The singularity at $K3$0 is removable, but the coordinates degenerate at $K3$1, the separated solution contains a term proportional to $K3$2, and a global extension would require controlling monodromy under coordinate transitions; the author does not pursue this and conjectures that no global nondegenerate solution exists.

The main rigidity theorem substantiates this conjecture under a quantitative hypothesis: there is no global critical $K3$3 harmonic function on $K3$4 branched along a planar hyperbola whose Almgren frequency $K3$5 has a finite limit at infinity. The proof is a scale-down (blow-up) argument. Rescaling $K3$6 by $K3$7 sends the hyperbola to a confocal hyperbola converging to two transverse lines through the origin—the branching set of the Taubes–Wu tetrahedral model, proved nondegenerate by Chen–He. The argument proceeds through:

• a monotonicity formula for $K3$8, established for critical $K3$9 harmonic functions by a stress-energy tensor computation, with boundary terms vanishing by the $\Sigma$0 asymptotics;
• a Caccioppoli inequality with logarithmic cut-offs adapted to the codimension-two singular set;
• extraction of a subsequence of rescalings along which $\Sigma$1 (using $\Sigma$2), giving uniform $\Sigma$3 bounds on the sphere $\Sigma$4;
• a bundle-matching lemma: although the rescaled functions live on distinct line bundles, the punctured spheres carry isomorphic flat bundles (same Stiefel–Whitney class), so all sections can be pulled back to a common reference bundle via diffeomorphisms converging to the identity;
• a Bochner-type $\Sigma$5 estimate showing the weak limit $\Sigma$6 lies in $\Sigma$7; a nonzero leading coefficient $\Sigma$8 in the expansion of the homogeneous extension would force $\Sigma$9 to blow up like $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$0 near the puncture, contradicting $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$1. Hence $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$2 is critical, and it is an eigensection on the sphere with eigenvalue $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$3.

The limit configuration $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$4 contains an antipodal pair, and Chen–He's rigidity theorem states that such configurations are non-critical—contradiction. The author is explicit that the bundle-matching step requires the branching sets to be mutually diffeomorphic; handling more general topological changes of the branching set is left open, and the finite-frequency hypothesis is essential to the compactness.

Global parabolic construction

In contrast to the hyperbolic case, a global construction exists for parabolic branching sets. Using modified paraboloidal coordinates (confocal paraboloids with $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$5, $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$6), the parabola $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$7 becomes the coordinate degeneracy locus $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$8, and again $$f=\operatorname{Re}\left(A\zeta^{1/2}+B\zeta^{3/2}\right)-\tfrac12\operatorname{Re}(\overline\mu\zeta)\operatorname{Re}(A\zeta^{1/2})+O(|\zeta|^{5/2})$$9. Separation of variables and an explicit reduction-of-order construction give

$A\equiv 0$0

where $A\equiv 0$1 are elementary integrals and $A\equiv 0$2 is a quadratic harmonic polynomial. The leading term is $A\equiv 0$3, so $A\equiv 0$4 is nondegenerate with branching set exactly $A\equiv 0$5. Unlike the hyperbolic case, no coordinate singularity obstructs the global extension.

The asymptotics at infinity are computed precisely: for $A\equiv 0$6 and large $A\equiv 0$7,

$A\equiv 0$8

derived from the closed forms of the three integrals and the implicit-function-theorem asymptotics of the largest paraboloidal coordinate root. The scale-down limit $A\equiv 0$9 converges in $M$00 on either sheet, with error $M$01, to $M$02 where $M$03. The limiting branching object is the ray $M$04 with multiplicity two; since $M$05 is contractible, the limit is an ordinary harmonic function vanishing on $M$06 rather than a genuine $M$07 harmonic function—a structural contrast with the hyperbolic scale-down, whose limit is a singular branched configuration.

The special Lagrangian interpretation

The parabolic model has a precise geometric origin. Joyce's affine-quadric special Lagrangians in $M$08, with parameters $M$09, $M$10, $M$11 and $M$12, yield a family $M$13 whose real projection is an exact two-valued Lagrangian graph over $M$14 branched along $M$15—independently of $M$16, which was the point of the normalization $M$17. The critical set of the projection is $M$18, mapping onto $M$19; for $M$20 the full map remains an embedding there, the missing rank of $M$21 being supplied by the imaginary directions.

Three facts are established. First, the central member $M$22 is parametrized by exactly the modified paraboloidal coordinates used to construct $M$23, with matching deck transformation. Second, since $M$24 is simply connected, the Liouville form is exact on it, giving an anti-invariant graph potential $M$25 vanishing on $M$26, computed explicitly in closed form. Third, the exact identity

$M$27

implies $M$28 in $M$29 on the cover. Thus $M$30 is the infinitesimal two-valued graph potential of Joyce's family: the linearization of the special Lagrangian equation $M$31 at $M$32 recovers harmonicity, and the asymptotic planes of $M$33 have potentials whose first $M$34-derivative is $M$35, consistent with the scale-down limit. This parallels the known interpretation of Yan's ellipsoidal models via Lawlor necks, and gives the parabolic model the same standing as a linear model for branched special Lagrangian deformations.

Limitations and open questions

The paper states its limitations plainly. The hyperbolic construction is local only; whether a global extension exists is conjectured negatively but proved only under the finite-frequency hypothesis, and the obstruction argument relies on the Chen–He rigidity theorem for configurations containing an antipodal pair, so it does not exclude global solutions with superlinear frequency growth. The bundle-matching technique in the compactness argument presupposes diffeomorphic branching sets, and the author notes that topological changes in the branching set remain unresolved. The hyperbolic and parabolic constructions are carried out only in dimension three; higher-dimensional analogues are expected to follow by the same methods but are not written out. Finally, the author observes a formal similarity between the local hyperbolic solution and a local special Lagrangian in Joyce's work but does not establish a direct geometric correspondence analogous to the parabolic case.

Conclusion

The paper completes the parametrization of Yan's ellipsoidal models by proving the injectivity of the coefficient map $M$36, establishes a conditional global rigidity result ruling out finite-frequency critical $M$37 harmonic functions branched along planar hyperbolas, and provides the first global nondegenerate example with a noncompact quadric branching set, the parabola, together with sharp asymptotics and a scale-down limit. The identification of the parabolic model with the infinitesimal potential of Joyce's special Lagrangian family gives the construction a concrete calibrated-geometry interpretation and suggests that explicit $M$38 models and special Lagrangian families are two descriptions of the same deformation theory in favorable cases.