First variation of area formula
Let 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
where dA is the area form on induced by the metric of M, and H is the mean curvature of . The normal vector is parallel to where is the tangent vector. The mean curvature is parallel to the normal vector.
- Chow, Lu, and Ni, "Hamilton's Ricci Flow." AMS Science Press, GSM volume 77, 2006.