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 before, but I did not get an answer. Thank you very much.


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.


The analyticity of Ricci solitons (not necessarily gradient) was first obtained, up to my knowledge, by Thomas Ivey in the paper Local existence of Ricci solitons, published in $1996.$ (see http://link.springer.com/article/10.1007%2FBF02567946?LI=true). You can get the paper following the link http://iveyt.people.cofc.edu/papers/locnew.pdf


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