Cognate linkage

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 $\mathrm{\Sigma }(t)$ 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

$\frac{d}{dt}dA=HdA,$

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