Lorentzian Conformal Flow in 3-Manifolds

• Lorentzian conformal flow is the evolution generated by a timelike conformal vector field that rescales the metric and connects contact and cosymplectic geometries in 3-manifolds.
• A conformal rescaling transforms the vector field into a unit-length Killing field, establishing a stable Hamiltonian structure and a rigid foliation with distinct topological constraints.
• The flow’s interplay with Sasakian and co-Kähler structures informs geometric evolution, reveals integrability in higher dimensions, and raises questions on stability under Cotton flow dynamics.

A Lorentzian conformal flow refers to the evolution generated by a conformal vector field on a Lorentzian manifold. In the context of three-dimensional closed Lorentzian manifolds, such flows exhibit remarkable rigidity and deep connections to stable Hamiltonian structures, contact and cosymplectic geometry, and the topology of flows. Their study intersects geometric analysis, foliation theory, and the theory of parabolic flows, reflecting both local and global structural phenomena.

1. Definition and Characterization of Lorentzian Conformal Flows

Let $(M,g)$ be a closed oriented 3-manifold with a Lorentzian metric of signature $(-,+,+)$. A smooth vector field $X\in\Gamma(TM)$ is called a conformal Killing vector field (conformal vector field) if its flow preserves the metric $g$ up to scale, i.e., there exists a smooth function $\sigma:M\to\mathbb{R}$ such that

$\mathcal{L}_X g = 2\sigma g,$

where $\mathcal{L}_X$ is the Lie derivative along $X$. $X$ is said to be timelike if $g(X,X)<0$ everywhere, and the function $\sigma$ is called the conformal factor or potential. The flow generated by such $X$—the Lorentzian conformal flow—infinitesimally rescales lengths by $e^{\sigma t}$, preserving the conformal structure but generally not the metric itself.

A crucial property is that on any neighborhood where $X$ is nowhere-vanishing, it can always be made Killing (i.e., $\sigma=0$ for some conformally equivalent metric), a result that underpins the local-inessentiality phenomenon in Lorentzian Ferrand-Obata theory (Dumitrescu et al., 5 Nov 2025).

2. Conformal Change and the Emergence of Stable Hamiltonian Structures

Any nowhere-vanishing timelike conformal vector field $X$ can be made unit-length and Killing by a conformal rescaling: set $\tilde g = e^{-2f}g$, with $f$ solving $Xf=\sigma$ and $e^{-2f} = -1/g(X,X)$. With respect to $\tilde g$, $X$ satisfies $\mathcal{L}_X \tilde g=0$ and $\tilde g(X,X) = -1$, so $X$ becomes a unit-length Killing field.

Passing to a compatible Riemannian metric $g_R = \tilde g + 2\alpha\otimes\alpha$ (where $\alpha = \tilde g(X,\cdot)$), one preserves the Killing property and can invoke Wadsley’s theorem to arrange $\nabla^R_X X=0$. The associated 1-form $\theta = \iota_X \operatorname{vol}_{g_R}$ and closed 2-form $\Omega = \iota_X(\operatorname{vol}_{g_R})$ generate a stable Hamiltonian structure (SHS): $\Omega$ is closed with kernel spanned by $X$, $\theta \wedge \Omega > 0$ is a volume form, and $d\theta = \tau\Omega$ for some $\tau$ constant along the flow of $X$. The Reeb field $R$ of this SHS coincides with $X$.

The dynamics generated by $R$ define a 1-dimensional Riemannian foliation $\mathcal{F}$ whose basic cohomology satisfies $\dim H^2_B(\mathcal{F}) \leq 1$ (Gnandi et al., 31 Dec 2025).

3. Classification Theorem: Sasakian and Co-Kähler Structures

The main classification result asserts that every nowhere-vanishing timelike conformal vector field on a closed Lorentzian 3-manifold, after a suitable conformal change, becomes the Reeb field of either a Sasakian or a co-Kähler structure:

• Sasakian case: There exists a contact 1-form $\eta$ with $\eta\wedge d\eta \neq 0$, a Reeb field $R$ satisfying $\eta(R)=1$ and $\iota_R d\eta = 0$, and a compatible normal almost contact structure; the induced metric is Riemannian with $R$ Killing. In three dimensions, every $K$-contact structure is automatically Sasakian. Sasakian 3-manifolds are Seifert fibered, the first Betti number $b_1$ is even (0 or 2), and there are at least two closed Reeb orbits.
• Co-Kähler (K-cosymplectic) case: There exists a pair $(\eta,\Phi)$ with $d\eta=0$, $d\Phi=0$, $\eta\wedge\Phi^n\neq0$, and a normal almost contact structure with Killing Reeb field. Compact co-Kähler 3-manifolds are mapping tori of Hermitian isometries of compact Kähler surfaces; $b_1$ is odd, and after finite covering, the manifold is $\Sigma\times S^1$ (Gnandi et al., 31 Dec 2025).

This splitting corresponds to whether the cohomological class $[d\theta]$ vanishes in the basic cohomology or not. The Reeb-like character of Lorentzian conformal flows enforces either contact or cosymplectic geometry, rigidifying the underlying manifold's topology and dynamics.

4. Local Theory and the Ferrand–Obata Theorem

In the local, real-analytic setting, the Lorentzian Ferrand–Obata theorem provides a dichotomy: any conformal vector field on a closed, real-analytic Lorentzian 3-manifold is locally Killing (after a suitable conformal change) except in the case where the manifold is conformally flat (Dumitrescu et al., 5 Nov 2025). If $X$ has a singularity with nonzero conformal distortion, the metric is globally conformally flat.

The classification of singularities (isometry-like, contracting/expanding, mixed, balanced) identifies where the flatness is forced and where a local Killing representative exists. In dimension three, the vanishing of the Cotton tensor (computed from the conformal vector field) is equivalent to conformal flatness; thus, local degeneracy directly links to the geometric structure of the manifold.

5. Lorentzian Cotton Flow and Geometric Evolution

The Cotton flow is a geometric evolution equation driving a 3-manifold's metric toward conformal flatness: $\partial_t g_{ij} = C_{ij}$ where $C_{ij}$ is the symmetric $2$-tensor version of the Cotton tensor, defined as

$$C_{abc} = \nabla_a R_{bc} - \nabla_b R_{ac} - \frac{1}{4} (g_{bc}\nabla_a R - g_{ac}\nabla_b R).$$

This construction is signature-independent and valid for Lorentzian metrics. The flow, when augmented by a DeTurck term and Weyl rescalings, is not strictly parabolic or hyperbolic in the Lorentzian category, and preserves neither causal nor conformal structure in a straightforward sense.

Linear stability analysis of Lorentzian Cotton flow about Einstein backgrounds reveals generically unstable modes with saddle-point behavior, implying that neither flat metrics nor generic Einstein metrics are stable fixed points under the flow. The existence of growing modes complicates its interpretation as a smoothing flow and raises open questions about global existence and singularity formation (Kilicarslan et al., 2015).

Cotton solitons—a metric + vector field pair solving $C_{ij} + \nabla_{(i}V_{j)} + \Phi g_{ij} = 0$—include non-trivial examples such as Lorentzian pp-waves, which are both Cotton and Ricci solitons. This connects Lorentzian conformal flows to topologically massive gravity and AdS/CFT phenomena.

6. Integrable Lorentzian Conformal Flows in Higher Dimensions

In four-dimensional Lorentzian geometry, the theory of conformal flows of star-shaped curves in the light cone provides a distinct domain. The differential invariants of such curves, under the action of the Lorentz group, are Poisson-equivalent to conformal invariants on the Möbius sphere. This equivalence allows for the realization of completely integrable systems (such as the complexly coupled KdV equations) as geometric flows of curves in the light cone: $$\begin{cases} \kappa_{1,t} = -\kappa_1''' + 3\kappa_1 \kappa_1' + 3\kappa_2 \kappa_2' \ \kappa_{2,t} = -\kappa_2''' + 3\kappa_1 \kappa_2' + 3\kappa_2 \kappa_1' \end{cases}$$ with the invariants $(\kappa_1,\kappa_2)$ corresponding to the geometric flow. Zero-curvature representations guarantee complete integrability (Anderson et al., 2017).

7. Geometric and Topological Implications

The identification of Lorentzian conformal flows with Reeb flows of stable Hamiltonian structures in dimension three strictly constrains the geometry and topology of the underlying manifold. In particular:

• Every nowhere-vanishing timelike conformal flow is intrinsically Reeb-like, enforcing either Sasakian (contact) or co-Kähler (cosymplectic) geometry after conformal rescaling.
• Topological consequences include constraints on the first Betti number and the structure of the foliation by the flow, governing the possible types of closed orbits and possible fibered structures.
• In the context of geometric flows, Lorentzian Cotton flow highlights the unique role of the Cotton tensor in driving conformal smoothing and identifies the boundaries imposed by linear instabilities and flat singularities.

This rigidity differentiates the Lorentzian case from its Riemannian counterpart and underscores the deep relationships between Lorentzian geometry, foliation theory, and odd-dimensional analogues of Kähler geometry (Gnandi et al., 31 Dec 2025, Dumitrescu et al., 5 Nov 2025, Kilicarslan et al., 2015, Anderson et al., 2017).