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.