# Thread: Metrics close implies volume forms close

1. ## 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 $\vert Area(S^2,g) - Area(S^2,\gamma) \vert \leq C \varepsilon$?

2. ## Re: Metrics close implies volume forms close

The Bishop-Gromov comparison theorem pops into mind.