Euler Lagrange Equation and Besse Conjecture

Question

Many paper said that Einstein Hilbert functional $E(g)$ defined as follows $$E(g) = \int_{M} R_{g}dM_{g}$$ If it restricted on unit volume. The Euler Lagrange can be writen as $$Ric - \frac{R}{n} = Hess(f) - (Ric - \frac{R}{n-1}g)f$$ for some function $f$. But how to compute the Euler Lagrange equation for this problem? I have doubt that and the reference always tell us it it easy to see that. But I really do not know. Is there any one know or recommand good references?

Answer 1

It has been discussed in some books and papers. For instance see

• Schoen, Richard M., Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations, Lect. 2nd Sess., Montecatini/Italy 1987, Lect. Notes Math. 1365, 120-154 (1989). ZBL0702.49038.
• Barros, Abdênago; Evangelista, Israel, On the critical metrics of the total scalar curvature functional, Publ. Math. 92, No. 1-2, 147-158 (2018). ZBL1399.53078.
• Viaclovsky, Jeff A., Critical metrics for Riemannian curvature functionals, Bray, Hubert L. (ed.) et al., Geometric analysis. Lecture notes from the graduate minicourse of the 2013 IAS/Park City Mathematics Institute session on geometric analysis, Park City, UT, USA, 2013. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study (IAS) (ISBN 978-1-4704-2313-1/hbk; 978-1-4704-2881-5/ebook). IAS/Park City Mathematics Series 22, 197-274 (2016). ZBL1360.53005.
• Besse, Arthur L., Einstein manifolds, Classics in Mathematics. Berlin: Springer (ISBN 978-3-540-74120-6/pbk). xii, 516 p. (2008). ZBL1147.53001.