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Math Help - Metrics close implies volume forms close

  1. #1
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    Metrics close implies volume forms close

    Let g be a metric on the S^2 which is close to the standard metric \gamma, i.e. \sup_{\theta \in S^2}\vert g - \gamma \vert \leq \varepsilon for some \varepsilon small, where \vert \cdot \vert is the norm with respect to \gamma (say). Is there an easy way of showing that the volume forms are close? Or even \vert Area(S^2,g) - Area(S^2,\gamma) \vert \leq C \varepsilon?
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Metrics close implies volume forms close

    The Bishop-Gromov comparison theorem pops into mind.
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