Let $ M $ be a Riemannian manifold and let $ f $ be an harmonic function on $ M $.
By Unique continuation theorem we can assert that if $ \nabla f = 0 $ on an open subset $ \Omega \subset M $ then $ f $ is constant on $ M $.
Now consider a non constant harmonic function $ f $. What can we say about the set of zeros of $ \nabla f $?
By what i say above this set has empty interior surely. Now it is natural to ask if this set has a zero measure. Is it true?
Thanks