Simons' formula explained

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 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, is the second fundamental form, is the mean curvature, and is the symmetric 2-tensor on given by .This has the consequence that(1)/(2) \Delta|h| ^{2=|\nabla} h| ^{2-|h|} ^{4+\langle} h,\operatorname{Hess}H\rangle + H\operatorname{tr}(A ^{3)} where 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 ^{p} _{i\nabla} h _{jp} - {{R ^{p}} _{ij} }^qh_-^qh_\\&=\nabla_i\nabla_jH-(h^h_-h_j^ph_i^q)h_-(h^h_-Hh_i^q)h_\\&=\nabla_i\nabla_jH-|h|^2h+Hh^2;\endthe 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.

References

FootnotesBooks

Articles