# A 2-form flow II

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Let $(M,g,J)$ be a hermitian manifold with 2-form $\omega (X,Y)=g(JX,Y)$. I define a 2-form flow:

$$\frac{\partial \omega}{\partial t}(X,Y)= \sum_i R(X,Y,e_i,Je_i)$$

where $R$ is the Riemann curvature and $(e_i)$ is an orthonormal basis.

Can we find solutions of the 2-form flow for short time?

asked Jun 4, 2022
edited Jun 4, 2022

We could also take:

$$\frac{\partial \omega}{\partial t}=\omega^* R$$

where $\omega^* R$ is the contraction of the Riemann curvature $R\in \Lambda^2 (TM) \otimes \Lambda^2(TM)$ by the 2-form $\omega$.

You need to impose some smoothness condition on the initial data to formulate a well-posed Cauchy problem, but qualitatively, yes. Another qualitative comment: the equation you have written down is _related_ to the Ricci flow equation used in the Hamilton-Perelman geometrization theorem, as it is closely related to the fundamental equation of the Helmholtz operator induced by the Riemannian structure.

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