Nonvanishing Harmonic 1-Forms

• Nonvanishing harmonic 1-forms are smooth differential forms that are both closed and co-closed, with a strictly positive norm across the manifold.
• They critically link geometric curvature, spectral theory, and topological invariants, serving as measurable indicators in classical Hodge theory and beyond.
• Recent studies, including neural network approaches and gluing techniques, have expanded their construction in complex settings like branched covers and Calabi–Yau manifolds.

A nonvanishing harmonic 1-form on a smooth manifold is a differential 1-form $\omega$ solving the harmonic equation (i.e., it is both closed and co-closed, or equivalently satisfies $\Delta \omega = 0$) whose pointwise norm $|\omega|$ is nowhere zero on the manifold or prescribed domain. The existence, vanishing, or topological and analytical significance of such forms is deeply tied to the geometry of the underlying manifold, the analytic structure of the Laplacian, spectral theory, stability and curvature conditions, as well as algebraic-topological invariants. The paper of nonvanishing harmonic 1-forms is central both in classical Hodge theory on compact manifolds and in modern geometric analysis on noncompact, singular, or branched settings.

1. Definitions and Background

A harmonic 1-form $\omega$ on a Riemannian manifold $(M,g)$ is a smooth 1-form satisfying

$d\omega = 0, \quad \delta\omega = 0,$

where $d$ is the exterior derivative and $\delta$ is the codifferential (the adjoint of $d$ with respect to the $L^2$ inner product defined by $g$). Equivalently, $\omega$ is a solution to the Hodge Laplacian: $\Delta \omega = (d\delta + \delta d)\omega = 0.$ On a manifold with measure $e^{-f} dv_g$ (a smooth metric measure space), the Laplacian and inner products are modified accordingly (e.g., introducing the $f$-Laplacian $\Delta_f$ and the $L^2_f$-norm).

A nonvanishing harmonic 1-form is a harmonic 1-form $\omega$ such that $|\omega(x)| > 0$ for all $x$ in the domain of interest. In more general settings such as branched covers or with local systems, "nonvanishing" may be interpreted away from a branching set or in the sense of nowhere vanishing sections of a twisted bundle.

2. Vanishing Theorems and Rigidity: Obstacles to Existence

Many geometric contexts preclude the existence of nonvanishing harmonic 1-forms by analytic or topological rigidity, often linked to curvature and spectral constraints.

2.1 Curvature and Spectral Gap Conditions

In "L^2 harmonic 1-forms on minimal submanifolds in hyperbolic space" (1007.0663), it is shown that if $M^n$ is a complete super stable minimal submanifold in hyperbolic space and its first Laplacian eigenvalue satisfies

$\lambda_1(M) > (2n-1)(n-1),$

then every $L^2$ harmonic 1-form is trivial. The proof leverages the Bochner formula and precise eigenvalue estimates, leading to spectral gap rigidity: under these analytic conditions, the only $L^2$ harmonic 1-form is the zero form.

Similarly, on smooth metric measure spaces $(M^n, g, e^{-f} dv)$ with m-Bakry–Émery Ricci curvature bounded below by $-\alpha \lambda_1(\Delta_f)$ for $0 < \alpha < \frac{m-1}{m}$, all $L^2_f$ harmonic 1-forms vanish (Zhou, 2020). In the borderline case, either harmonic 1-forms vanish, or the universal cover splits as $\mathbb{R} \times N$ ("splitting theorem").

This type of result extends to $L^p_f$ harmonic forms and to weighted Poincaré inequalities, establishing the near equivalence between vanishing of (square-integrable) harmonic 1-forms and lower curvature/spectral gap bounds (Zhou et al., 2020). These vanishing theorems generally imply strong restrictions on the topology—constraining the first Betti number and the structure of the space of closed forms modulo exact forms.

2.2 Special Holonomy, Kähler, and Almost Kähler Manifolds

On complete special holonomy manifolds, such as $G_2$- or $Spin(7)$-manifolds with $d$(linear) structure forms, every $L^2$ harmonic 1-form must vanish. The injectivity of the Lefschetz mapping and wedge product arguments, together with the d(linear) property and the Ricci-flatness imposed by the holonomy, force the triviality of low-degree $L^2$ harmonic forms (Huang, 2018).

Akin results hold for almost Kähler manifolds: vanishing theorems (extending those of Gromov, Jost–Zuo, Cao–Xavier) show that if the symplectic form is $d$(bounded) or $d$(sublinear), $L^2$ harmonic $(p, q)$-forms can be nonzero only if $p+q = n$ (middle degree), ruling out nontrivial $L^2$ harmonic 1-forms if $n > 1$ (Huang, 2021).

For Stein manifolds with $d$-bounded geometry, the space of $L^2$ Bott–Chern harmonic forms vanishes in all degrees except the middle ($p+q = n$), implying the absence of nonvanishing harmonic 1-forms for $n>1$ (Piovani et al., 2018).

2.3 Parallel 1-forms and Vaisman Geometry

If a complete Riemannian manifold admits a nonzero parallel 1-form (for example, a universal covering of a Vaisman manifold), every $L^2$ harmonic form vanishes identically (Huang et al., 2019). This is proved using the Morse-Novikov differential, the vanishing of appropriate cohomologies, and the Künneth formula for $L^2$-cohomology.

3. Existence: Classical Hodge Theory and Topological Criteria

On compact manifolds, classical Hodge theory asserts that the space of harmonic 1-forms is isomorphic to the first de Rham cohomology group: $$H^1(M;\mathbb{C}) \cong \{\omega \in \Omega^1(M) \mid \Delta \omega = 0 \}.$$ Thus, nontrivial $H^1(M;\mathbb{C})$ implies the existence of (generally nonvanishing) harmonic 1-forms. On compact Kähler manifolds, the decomposition

$$H^1(M;\mathbb{C}) \cong H^{1,0}(M) \oplus H^{0,1}(M)$$

ensures that nonzero holomorphic or anti-holomorphic 1-forms arise as harmonic representatives (Lim, 2022).

The key point is that, on compact manifolds, the existence of nontrivial harmonic 1-forms reduces to the computation of topological invariants (e.g., Betti numbers). For example, on the $n$-torus $T^n$, there are $n$ linearly independent nonvanishing harmonic 1-forms.

4. Nonvanishing Harmonic 1-Forms in Singular, Branched, or Multi-Valued Settings

In more exotic settings, such as branched covers or with twisted coefficients, the notion of harmonic 1-forms must be generalized to account for multivaluedness, monodromy, or singular sets.

4.1 $\mathbb{Z}_2$- and Multi-Valued Harmonic 1-Forms

In "New examples of Z/2 harmonic 1-forms and their deformations" (Haydys et al., 2023) and "Existence of nondegenerate $\mathbb{Z}_2$ harmonic 1-forms via $\mathbb{Z}_3$ symmetry" (He, 2022), explicit constructions of $\mathbb{Z}_2$ harmonic 1-forms are provided. Such a form may be globally defined on a branched double cover but only defined up to sign on the underlying manifold, with a branching locus $\Sigma$ of codimension 2. The forms are harmonic away from $\Sigma$ and vanish along $\Sigma$ at controlled rates, ensuring continuity (or boundedness) across the branch.

A key innovation is the use of symmetry (e.g., $\mathbb{Z}_3$ actions) to force vanishing of certain singular terms in a local expansion, ensuring a bounded and nondegenerate (nowhere zero outside $\Sigma$) harmonic 1-form. The existence of such forms depends on both topological conditions (linking numbers, monodromy) and symmetry considerations.

These results imply the existence of nonvanishing (in the appropriate sense) harmonic 1-forms on large classes of branched spaces, including infinitely many rational homology spheres and manifolds constructed as cyclic covers over links with vanishing determinant.

4.2 Gluing and Abundance on Connected Sums

Recent advances (He et al., 15 Jul 2024) extend these constructions by gluing arguments (via the Nash–Moser implicit function theorem). It is shown that, for any closed 3-manifold $Y$, there exist infinitely many $\mathbb{Z}_2$-harmonic 1-forms with singular sets in distinct isotopy classes, and, when $b_1(Y) > 0$, infinitely many spin$^c$ structures for which the moduli space of solutions to Seiberg–Witten equations is noncompact. The gluing method assembles local model solutions and corrects them to global harmonic representatives, illustrating the abundance of such objects in three-dimensional topology.

5. Numerical and Algorithmic Methods: The Calabi–Yau Case

Recent work (Douglas et al., 29 May 2024) employs neural networks to address the existence of nowhere vanishing harmonic 1-forms on real loci of Calabi–Yau manifolds. The workflow involves:

• Training a neural network to approximate a Ricci-flat (Calabi–Yau) metric by minimizing a complex Monge–Ampère loss.
• Using a separate neural network to approximate an $L^2$-normalized, closed and co-closed 1-form (via minimization of the Hodge energy functional)
• Checking numerically whether the optimized 1-form is nowhere vanishing by evaluating the pointwise norm over sufficiently dense samples.

These methods lead to the discovery that, on certain real loci diffeomorphic to $S^1 \times S^2$, a nowhere vanishing harmonic 1-form likely exists (for example, in the "CICY2" case), while on loci diffeomorphic to $RP^3$ or $RP^3 \# (S^1\times S^2)$, nonexistence follows from topological constraints (e.g., vanishing first Betti number or the Poincaré–Hopf theorem). The neural network approach is amenable to rigorous, computer-assisted proofs if suitable a posteriori estimates can be established.

6. Analytical Structure and Regularity

The analytic paper of harmonic 1-forms often centers on elliptic regularity, singularity analysis, and the local structure near branching sets or singular submanifolds. In harmonic spaces (Gilkey et al., 2020), for instance, the singularity of (radial) harmonic 1-forms at a point is controlled via explicit ODE analysis, showing that up to two linearly independent solutions exist: one "regular" (nonvanishing) and one possibly logarithmically singular. The precise local expansion (involving, for example, terms like $z^{1/2} dz$ for the $\mathbb{Z}_2$ case) determines both the boundedness and the vanishing order along the singular set.

Near a branched set $\Sigma$, the asymptotic expansion for a multivalued harmonic 1-form $v$ reads

$$v \sim \mathrm{Re}\left\{ d\left(A(t) z^{1/2}\right) + d\left(B(t)z^{3/2}\right) \right\} + O(r^{3/2-}),$$

and boundedness (hence nonvanishing away from $\Sigma$) is ensured if $A(t)\equiv 0$ and $B(t)$ is nowhere zero (He, 2022).

7. Summary Table of Existence/Vanishing Criteria

Geometric Setting	Existence of Nonvanishing Harmonic 1-Forms	Reference
Compact Riemannian/Kähler manifolds with $b_1(M) > 0$	Yes: Via Hodge theory, dimension of the space is $b_1(M)$.	(Lim, 2022)
Complete minimal submanifold in $\mathbb{H}^{n+m}$, $\lambda_1 > (2n-1)(n-1)$	No: All $L^2$ harmonic 1-forms vanish.	(1007.0663)
Special holonomy ($G_2$/$Spin(7)$) w/ d(linear) structure	No: All $L^2$ harmonic 1-forms vanish.	(Huang, 2018)
Complete Riemannian manifold with parallel 1-form	No: All $L^2$ harmonic 1-forms vanish.	(Huang et al., 2019)
Noncompact Kähler/Stein, $d$-bounded, $n > 1$	No: $L^2$ harmonic 1-forms vanish except in middle degree.	(Piovani et al., 2018)
Real locus of some Calabi–Yau $L \cong S^1 \times S^2$	Yes (numerically): Neural network finds candidate nowhere vanishing form	(Douglas et al., 29 May 2024)
Branched/multivalued ($\mathbb{Z}_2$, $\mathbb{Z}_3$) case	Yes (away from singularity): Many explicit constructions.	(He, 2022, Haydys et al., 2023)
Gluing on 3-manifolds, $\mathbb{Z}_2$-harmonic forms	Yes: Infinitely many exist with distinct singular sets.	(He et al., 15 Jul 2024)

8. Open Problems and Future Directions

• Topological obstructions: Fine-grained classifications of when nonvanishing harmonic 1-forms exist, especially in noncompact, singular, or metric measure settings, are not complete. For real loci of Calabi–Yau manifolds, computational and theoretical methods help identify viable geometric types (e.g., $S^1 \times S^2$) (Douglas et al., 29 May 2024).
• Branched and multivalued settings: The connection between the topology of the branching set $\Sigma$, the structure of the twisted (or local system) cohomology, and the existence/nonvanishing nature of harmonic forms continues to generate new examples and theoretical questions (Haydys et al., 2023, He, 2022).
• Algorithmic and neural network methods: The use of data-driven architectures for explicit metrics and forms is a rapidly developing area, promising rigorous numerical existence proofs and deeper exploration of metric moduli spaces (Douglas et al., 29 May 2024).
• Gauge theory and Seiberg–Witten moduli: The abundance of nonvanishing (branched or twisted) harmonic 1-forms has implications for the compactness properties and structure of the Seiberg–Witten moduli spaces in low dimensions (He et al., 15 Jul 2024).

References to Key Results

• (1007.0663): Nonexistence of $L^2$ harmonic 1-forms on super stable minimal submanifolds in hyperbolic space with large spectral gap.
• (Huang, 2018): Vanishing results for $L^2$ harmonic 1-forms on complete special holonomy manifolds with $d$(linear) structure.
• (Piovani et al., 2018): Vanishing of Bott–Chern harmonic 1-forms on $d$-bounded Stein manifolds except in the middle degree.
• (Huang et al., 2019): Parallel 1-forms imply vanishing of $L^2$ harmonic forms; explicit on Vaisman manifolds.
• (Lim, 2022): Hodge theorem—existence of nonvanishing harmonic 1-forms equivalent to first Betti number nonzero.
• (Haydys et al., 2023): Explicit construction and deformation theory of $\mathbb{Z}_2$ harmonic 1-forms—branch loci can be very general.
• (He, 2022): $\mathbb{Z}_3$ symmetry enforces bounded, nondegenerate $\mathbb{Z}_2$ harmonic 1-forms via topological/symmetry conditions.
• (Douglas et al., 29 May 2024): Neural network computation of harmonic 1-forms on real loci of Calabi–Yau manifolds; existence linked to $L \cong S^1 \times S^2$.
• (He et al., 15 Jul 2024): Infinite abundance of nonvanishing (branched, multi-valued) $\mathbb{Z}_2$ harmonic 1-forms on 3-manifolds.

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In summary, the existence of nonvanishing harmonic 1-forms is governed by a complex interplay between geometry (curvature, holonomy), analysis (spectral theory, PDEs on forms), and topology (cohomology, branching data). While vanishing is enforced by spectral gap or holonomy rigidity in many settings, richer geometric and topological contexts—including singular, branched, or multivalued cases—admit a wide diversity of nonvanishing solutions, sometimes with strong constraints and sometimes in great abundance. The analytical and numerical techniques developed to address these problems have deepened both our theoretical understanding and practical capacity to construct and characterize such forms on a wide range of spaces.