Laplacian and differential topology

[JMessenger]

Hi all,

 

I am researching a hypothesis and looking for anyone who is familiar with differential topology (specifically Einstein manifolds). I have access to the Besse book Einstein Manifolds but am also looking for any open questions in differential topology that I am not aware of. I am attempting to develop a solid proof link between the Laplacian and Einstein manifolds (listed in the book as not found yet).

 

The math is pretty basic, sort of a gauge theory of the Laplacian. In graphical form (as the equations might be confusing without them)

 

there is no distinction between the derivatives of the following plots:

 

 

therefore for a higher dimension scalar field there won't be any distinction between gradients nor the divergences of the functions:

[math]\nabla^2f_1=\nabla^2(0-f_2)=\nabla^2(C-f_3)[/math].

 

If [math]\nabla^2(C-f_3)=0[/math] are components of a four dimensional scalar field and are normalized I should get a flat signature of [math]diag[1-1,1-1,1-1,1-1)[/math]. If the field has Lorentzian invariance then by setting the constant of integration portion to zero we get [math]diag[-1,1,1,1)[/math]. This would be the same as the link between Ricci manifold and EInstein manifold.

Edited January 31, 2013 by JMessenger

[ajb]

Is your basic desire to say something special about the Laplacian on Einstein Manifolds? You mean the Laplacian acting on functions? It may also be interesting to think about it acting on densities.

 

When I get some time I will have a think about this... but no promises.

[JMessenger]

Is your basic desire to say something special about the Laplacian on Einstein Manifolds? You mean the Laplacian acting on functions? It may also be interesting to think about it acting on densities.

 

When I get some time I will have a think about this... but no promises.

There is the known analogue between a Ricci flat manifold [math]R=0[/math] and the Laplacian [math]\nabla^2\Phi=0[/math] (both source free). My basic desire is to show a link between Einstein manifold [math]R=4\Lambda[/math] or [math]4\Lambda-R=0[/math] and a Laplacian of the form [math]\nabla^2(C-f)=0[/math] (would also be source free). It does also get into densities within stress-energy tensors later, which is even more interesting to me, but this would need to be proven first. Ok the real motive is that the geometers who write under the name Arthur Besse are offering a free meal to whom ever comes up with a solution heh. It would help bolster my case if I can find other known differential topology problems that this would solve. I have read somewhere that there is a Minkowski space problem, but can't find the specifics of what was meant by that.

Edited January 31, 2013 by JMessenger