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Let (M,g,J) be a hermitian manifold with 2-form ω(X,Y)=g(JX,Y). I define a 2-form flow:
∂ω∂t(X,Y)=∑iR(X,Y,ei,Jei)
where R is the Riemann curvature and (ei) is an orthonormal basis.
Can we find solutions of the 2-form flow for short time?
We could also take:
∂ω∂t=ω∗R
where ω∗R is the contraction of the Riemann curvature R∈Λ2(TM)⊗Λ2(TM) by the 2-form ω.
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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