I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons". 
A complete Riemannian manifold $(M^n,g\_{ij})$ is called a *gradient shrinking Ricci soliton* if there exists a smooth function $f$ such that the Ricci tensor $R\_{ij}$ of the metric $g\_{ij}$ satisfies $$R\_{ij}+\nabla\_i\nabla\_j f=\rho g\_{ij}$$
for some constant $\rho>0$. 


In the proof, it asserts that (P.1164): Suppose $|\nabla f|^2=0$ on some nonempty open set of $M$. Since any gradient shrinking Ricci soliton is analytic in harmonic coordinates, it follows that $|\nabla f|^2=0$ on $M$. 


I would like to understand why "any gradient shrinking Ricci soliton is analytic in harmonic coordinates". Also, I don't understand why "any gradient shrinking Ricci soliton is analytic in harmonic coordinates" would imply that $|\nabla f|^2=0$ on $M$. I have asked this question in [math.stackexchange](https://math.stackexchange.com/questions/730964/gradient-ricci-soliton) before, but I did not get an answer. Thank you very much.



Gradient Ricci soliton
A sketch of a proof of the real analyticity of the metric and potential in harmonic coordinates for the related gradient Einstein solitons can be found in the paper <http://cvgmt.sns.it/media/doc/paper/2197/GRADIENT@EINSTEIN@SHRINKERS_24gen.pdf> on page 5 and 6. Once you know that f is real analyic, it follows by analytic continuation that the vanishing of the derivative $\nabla f$ on an open set $U\subset M$ implies that $\nabla f=0$ on all of $M,$ provided $M$ is connected. 


As is often the case with regularity issues for Elliptic equations, this proof ushers you back to a theorem in the epic book of Morrey "Multiple integrals in the calculus of variations." 


I should say that I'm nothing of an expert on these analytical issues, I just "know where to look." Perhaps someone with more expertise can give a more helpful answer.


