First variation of area formula

In Riemannian geometry, the first variation of area formula relates the mean curvature of a hypersurface to the rate of change of its area as it evolves in the outward normal direction.

Let $\Sigma(t)$ be a smooth family of oriented hypersurfaces in a Riemannian manifold M such that the velocity of each point is given by the outward unit normal at that point. The first variation of area formula is

\frac{d}{dt}\, dA = H \,dA,

where dA is the area form on $\Sigma(t)$ induced by the metric of M, and H is the mean curvature of $\Sigma(t)$ . The normal vector is parallel to $D_{\alpha} \vec{e}_{\beta}$ where $\vec{e}_{\beta}$ is the tangent vector. The mean curvature is parallel to the normal vector.

References

Chow, Lu, and Ni, "Hamilton's Ricci Flow." AMS Science Press, GSM volume 77, 2006.

This article is issued from Wikipedia - version of the 4/8/2011. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.