First variation of area formula

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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 Σ(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 Σ(t) induced by the metric of M, and H is the mean curvature of Σ(t).

[edit] References

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