Simons' formula

In the mathematical field of differential geometry, the Simons formula (also known as the Simons identity, and in some variants as the Simons inequality) 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 $M$ of Euclidean space, the formula asserts that
 * $$\Delta h=\operatorname{Hess}H+Hh^2-|h|^2h,$$

where, relative to a local choice of unit normal vector field, $h$ is the second fundamental form, $H$ is the mean curvature, and $h^{2}$ is the symmetric 2-tensor on $M$ given by $h2 ij = g^{pq}h_{ip}h_{qj}$. This has the consequence that
 * $$\frac{1}{2}\Delta|h|^2=|\nabla h|^2-|h|^4+\langle h,\operatorname{Hess}H\rangle+H\operatorname{tr}(A^3)$$

where $A$ is the shape operator. In this setting, the derivation is particularly simple:
 * $$\begin{align}

\Delta h_{ij}&=\nabla^p\nabla_p h_{ij}\\ &=\nabla^p\nabla_ih_{jp}\\ &=\nabla_i\nabla^p h_{jp}-{{R^p}_{ij}}^qh_{qp}-{{R^p}_{ip}}^qh_{jq}\\ &=\nabla_i\nabla_jH-(h^{pq}h_{ij}-h_j^ph_i^q)h_{qp}-(h^{pq}h_{ip}-Hh_i^q)h_{jq}\\ &=\nabla_i\nabla_jH-|h|^2h+Hh^2; \end{align}$$ the only tools involved are the Codazzi equation (equalities #2 and 4), the Gauss equation (equality #4), and the commutation identity for covariant differentiation (equality #3). 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.