Gauss curvature of $(M, f^2 g)$

Asked Dec 05 '20 at 09:15

Active Dec 05 '20 at 13:11

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When I am reading a book about minimal surface, there is a question show up.

Let $(M, g)$ be a Riemann surface, and $f$ be a positive smooth function on $M$, then the curvature of $(M, f^2 g)$ is given by $$K_g=\Delta_g\ln f+f^2K_{f^2g}\,.$$

I think it maybe a trivial question that just need to calculate carefully, but I have failed to get the answer. Is there any other way to obtain this result?

edited Dec 05 '20 at 13:11

asked Dec 05 '20 at 09:15

ling

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What do you mean by "another way"? What way have you tried? – Jack Lee Dec 05 '20 at 16:03
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If you search, you will find this question several times. Here is one solution, using moving frames; I did catch a missing term and edited later. – Ted Shifrin Dec 06 '20 at 00:35

Gauss curvature of $(M, f^2 g)$

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