Simons' formula
In the mathematical field of differential geometry, the Simons formula is a fundamental equation in the study of minimal submanifolds. It was discovered by James Simons in 1968. It can be viewed as a formula for the Laplacian of the second fundamental form of a Riemannian submanifold. It is often quoted and used in the less precise form of a formula or inequality for the Laplacian of the length of the second fundamental form.
In the case of a hypersurface of Euclidean space, the formula asserts that
where, relative to a local choice of unit normal vector field, is the second fundamental form, is the mean curvature, and is the symmetric 2-tensor on given by.
This has the consequence that
where is the shape operator. In this setting, the derivation is particularly simple:
the only tools involved are the Codazzi equation, the Gauss equation, and the commutation identity for covariant differentiation. The more general case of a hypersurface in a Riemannian manifold requires additional terms to do with the Riemann curvature tensor. In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.