Suppose f:R^3\(Ball of radius 1)--->R is smooth and satisfies f(S^2)=0, ie the unit sphere is a level set of f. does it neccessarily follow that f is a spherically symmetric function?
Yes i realised as soon as i went to bed that there are many counterexamples. it was just a hopeful leap on my part-i am trying to prove the equality case of the penrose inequality for graphs over R^3. Basically, i showed in my proof of the main part of the theorem that equality holds only if the domain of f isnt defined on some ball. i was then wondering if i could use this fact and birkhoffs theorem to conclude that the graph is in fact isometric to schwarzschild. thanks for your replies anyhow