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Math Help - spherically symmetric function

  1. #1
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    spherically 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?


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  2. #2
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    Re: spherically symmetric function

    No. Why do you think so?
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  3. #3
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    Re: 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.
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  4. #4
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    Re: 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
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