## Approriate scalling in Yamabe equation

Suppose is a Riemannian manifold, and is the conformal Laplacian. Assume satisfies

where is some fixed constant, . Suppose near a point , there is a coordinates . We want to scale the coordinates to ,

By the conformal invariance of , for any , we get

We want to choose such that

which means

Letting

we get .

The above proof may not be right.

Or we should look it more directly

then

with .

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