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Thread: Metrics close implies volume forms close

  1. #1
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    Metrics close implies volume forms close

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