Happy New Year 2015!

In the last entry in 2014, I talk about conformal change of the Laplace-Beltrami operator. Given a Riemannian manifold of dimension . We denote a conformal metric of where the function is smooth.

Recall the following formula for the Laplace-Beltrami operator calculated with respect to the metric :

where is the determinant of . Then, it is natural to consider the relation between and in terms of . Recall that by we mean, in local coordinates, the following

hence by taking the inverse, we obtain

Clearly,

hence

With all these ingredient, we easily obtain

Thus, conformal change of the Laplace-Beltrami operator follows the following rule

When , we simple have

Keep in mind that can be calculated using ; hence one can use conformal changes for and to derive the above formula.

In local coordinates,

and

See also:

- Conformal Changes of the Green function for the conformal Laplacian.
- Conformal Changes of Riemannian Metrics
- Why should we call ” ” conformal change?

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Hi,

I am not a professional Mathematician but interested in Mathematics.

When I read the following notes about Laplacian:

Click to access Laplacian.pdf

I found difficulty in the exercises in P.45, about conformal transformations.

So , I google and found your useful page and this link:

http://math.stackexchange.com/questions/1071506/conformal-transformation-of-the-divergence

I have the same calculation as yours but this is mismatch with the exercise.

Can you spare some time to help me find out what’s wrong?

Thanks a lot.

Comment by CW CHU — March 3, 2015 @ 8:08

Hi, thanks for your interest in my blog and the comment.

In “4.3 The Laplacian under conformal deformations (Exercise)”, they use the conformal transformation ; hence to derive the 3rd identity, we let . An other remark is that they define .

3. Hence, from my formula

and thanks to

we obtain

Finally, using their notion, we can write

hence

as claimed. Note that, there is a typo in their note, you may check with Wiki.

Comment by Ngô Quốc Anh — March 3, 2015 @ 11:09

Thank you, I found the same typo in the notes of Yaiza Canzani and this post clarified the thing very well.

Comment by negropeppe — August 1, 2019 @ 20:23

Thank you very much!

“typo”, oh, it cost me a lot of time!

I find that your blog is useful for me (my interest is PDE and geometry). I will bookmark your blog and check it frequently.

Comment by CW CHU — March 3, 2015 @ 13:10

[…] of radius . Under conformal changes to the metric on a general riemannian manifold it is well-known that the Laplacian changes as . In particular, scaling the radius from 1 m to 3246 Mpc. will […]

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