Gradient of spectral function on noncompact homogeneous space

Let (M,g) be a noncompact Riemannian manifold whose isometry group acts transitively on M, i.e. a (not necessarily normal) homogeneous space. Let eλ(x,y) be the integral kernel of
fλ0dEν(f) where dEν is the spectral measure of the (non-negative) Laplacian associated to (M,g). Is there a relatively simple proof that

M|xeλ(x,y)|2gdy=Meλ(x,y)Δxeλ(x,y)dy ?

Note that the differentiation is in x and the integration is in y.
In particular, it’s surely false for most non-homogeneous manifolds.

I ask for `relatively simple’ because I know of a proof using the expression of the Laplacian as the generator of the heat semi-group. That proof seems to be overkill.


Source : Link , Question Author : Chris Judge , Answer Author : Community

Leave a Comment