We propose a new geometric flow that simultaneously evolves both the Riemannian metric and almost-complex structure of a manifold, driving the geometry toward a Kähler configuration. This flow couples the classical Ricci flow with a Nijenhuis torsion annihilation mechanism, potentially providing a new approach to the existence problem for Kähler metrics.
Let $(M, g_0, J_0)$ be a smooth manifold equipped with:
Definition 1.1 (Nijenhuis Tensor): The Nijenhuis tensor $N_J$ measures the failure of $J$ to be integrable:
$$N_J(X,Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X,Y]$$
By the Newlander-Nirenberg theorem, $N_J \equiv 0$ if and only if $J$ is integrable (comes from a complex structure).
Definition 1.2 (Kähler Manifold): $(M, g, J)$ is Kähler if:
Definition 2.1 (Ricci-Nijenhuis Flow): We define the coupled system:
$$\begin{cases} \frac{\partial g}{\partial t} = -2\text{Ric}(g) - Q(N_J, N_J) \[8pt] \frac{\partial J}{\partial t} = -\nabla^* N_J \end{cases}$$
Where:
Definition 2.2 (Energy Functional): Define the total geometric energy:
$$E[g, J] = \int_M \left( R^2 + |N_J|_g^2 \right) dV_g$$
Where $R$ is the scalar curvature and $|\cdot|_g$ denotes the norm induced by $g$.
The Ricci-Nijenhuis flow is the gradient flow of the energy functional $E[g,J]$ with respect to an appropriate geometric structure on the space of pairs $(g,J)$.
Specifically, we conjecture:
$$\frac{dE}{dt} = -\int_M \left| \frac{\partial g}{\partial t} \right|^2 + \left| \frac{\partial J}{\partial t} \right|^2 , dV_g \leq 0$$
Physical Interpretation: Energy dissipates monotonically along flow lines, suggesting the system relaxes toward equilibrium states.
For any compact manifold $(M, g_0, J_0)$ with $g_0$ complete and $J_0$ smooth, there exists $T > 0$ such that the Ricci-Nijenhuis flow has a unique smooth solution $(g(t), J(t))$ for $t \in [0,T)$.
Remark: This would follow from standard parabolic PDE theory if the system can be shown to be strictly parabolic with appropriate symbol structure.
Let $(M, g(t), J(t))$ be a solution to the Ricci-Nijenhuis flow on a compact manifold. If:
Then as $t \to \infty$:
$$|N_{J(t)}|{L^2} \to 0 \quad \text{and} \quad g(t) \to g\infty$$
Where $(M, g_\infty, J_\infty)$ is a Kähler manifold with $N_{J_\infty} = 0$.
Significance: This would provide a flow-based approach to constructing Kähler metrics, analogous to how Ricci flow approaches Einstein metrics.
There exists a geometric quantity $\mathcal{W}[g, J]$ (analogous to Perelman's $\mathcal{W}$-entropy for Ricci flow) such that:
$$\frac{d\mathcal{W}}{dt} \geq 0$$
with equality if and only if $(g,J)$ is Kähler.
Candidate: $$\mathcal{W}[g,J] = \int_M \left( R + |N_J|^2 \right) e^{-f} dV_g$$
for some auxiliary function $f$ evolving by:
$$\frac{\partial f}{\partial t} = -\Delta f + |\nabla f|^2 - R - |N_J|^2$$
When $J$ is held fixed and integrable ($N_J = 0$), the flow reduces to: $$\frac{\partial g}{\partial t} = -2\text{Ric}(g)$$
This is Hamilton's classical Ricci flow, for which extensive theory exists.
If we start with a Kähler metric and evolve only $g$: $$\frac{\partial g}{\partial t} = -2\text{Ric}(g), \quad J \text{ fixed}$$
This is the Kähler-Ricci flow, studied extensively by Cao, Hamilton, and Tian. However, their flow assumes $J$ is integrable a priori.
Key Difference: The Ricci-Nijenhuis flow evolves $J$ itself to achieve integrability, rather than assuming it.
Question 5.1: What is the explicit formula for $Q(N_J, N_J)$?
Natural candidate: $Q(N_J, N_J) = \frac{1}{2}g^{ik}g^{jl}N_{ij}^m N_{kl}^n g_{mn}$
Question 5.2: Can the flow develop finite-time singularities similar to Ricci flow?
Question 5.3: On which manifolds does the flow converge to a Kähler metric?
Question 5.4: Is there a Ricci-Nijenhuis flow analogue of Perelman's no local collapsing theorem?
Question 5.5: Can this flow be used to prove existence of Kähler metrics on manifolds where other methods fail?
Kähler manifolds (particularly Calabi-Yau) are central to string compactifications. A flow-based construction could provide new vacuum solutions.
The flow provides a computational algorithm: start with any $(g_0, J_0)$ and evolve toward Kähler geometry.
New monotonicity formulas could yield topological obstructions or classification results.
The flow has been implemented using a discrete time-stepping scheme:
Given: (M, g₀, J₀), time step dt
Repeat:
1. Compute Ric(g)
2. Compute N_J
3. Update: g ← g - dt(2Ric(g) + Q(N,N))
4. Update: J ← J - dt(∇*N_J)
5. Check: ||N_J|| < ε (convergence)Numerical Evidence: Preliminary simulations suggest:
This flow was discovered through automated mathematical reasoning and has not been rigorously studied. We seek:
Classical Results:
Related Flows:
This coupled flow emerged from an AI-driven exploration of geometric architectures. While inspired by classical results, the specific coupling appears novel. The conjectures are based on computational evidence and theoretical analogy with known flows. Rigorous proofs remain open problems.
We welcome feedback, counterexamples, or proof attempts from the mathematical community.
Keywords: Ricci flow, Nijenhuis tensor, Kähler manifolds, geometric flows, almost-complex structures, parabolic PDEs
MSC 2020: 53E20 (Flows related to mean curvature), 53C55 (Global differential geometry of Hermitian and Kählerian manifolds), 35K55 (Nonlinear parabolic equations)