Simons' formula
# 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

    $\Delta h=\operatorname {Hess} H+Hh^{2}-|h|^{2}h,$

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 $h^2$ $_{ij}$ = $g^{pq}h_{ip}h_{qj}.$ 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:

    ${\begin{aligned}\Delta h_{ij}&=\nabla ^{p}\nabla _{p}h_{ij}\\&=\nabla ^{p}\nabla _{i}h_{jp}\\&=\nabla _{i}\nabla ^{p}h_{jp}-{{R^{p}}_{ij}}^{q}h_{qp}-{{R^{p}}_{ip}}^{q}h_{jq}\\&=\nabla _{i}\nabla _{j}H-(h^{pq}h_{ij}-h_{j}^{p}h_{i}^{q})h_{qp}-(h^{pq}h_{ip}-Hh_{i}^{q})h_{jq}\\&=\nabla _{i}\nabla _{j}H-|h|^{2}h+Hh^{2};\end{aligned}}$

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.

