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?

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- Jun 4th 2013, 04:06 PMtommyjbabyspherically symmetric function
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?

- Jun 5th 2013, 12:12 PMxxp9Re: spherically symmetric function
No. Why do you think so?

- Jun 5th 2013, 12:40 PMHallsofIvyRe: spherically symmetric function
You said "the unit sphere is

**a**level set of f". (Emphasis mine). It does not follow that there are not other, non-symmetrical level sets. - Jun 5th 2013, 12:47 PMtommyjbabyRe: 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