Simons' formula

Template:Short description 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.Template:Sfnm 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 Template:Mvar of Euclidean space, the formula asserts that

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

where, relative to a local choice of unit normal vector field, Template:Mvar is the second fundamental form, Template:Mvar is the mean curvature, and Template:Math is the symmetric 2-tensor on Template:Mvar given by Template:Math.Template:Sfnm This has the consequence that

$\frac{1}{2}\mathrm{\Delta }|h{|}^2=|\mathrm{\nabla }h{|}^2-|h{|}^4+⟨h,\mathrm{Hess}H⟩+H\mathrm{tr}(A^3)$

where Template:Mvar is the shape operator.Template:Sfnm In this setting, the derivation is particularly simple:

$\begin{array}{cc}\mathrm{\Delta }{h}_{ij}& ={\mathrm{\nabla }}^p{\mathrm{\nabla }}_p{h}_{ij}\\ & ={\mathrm{\nabla }}^p{\mathrm{\nabla }}_i{h}_{jp}\\ & ={\mathrm{\nabla }}_i{\mathrm{\nabla }}^p{h}_{jp}-{{R^p}_{ij}}^q{h}_{qp}-{{R^p}_{ip}}^q{h}_{jq}\\ & ={\mathrm{\nabla }}_i{\mathrm{\nabla }}_{j}H-(h^{pq}{h}_{ij}-h_j^p{h}_i^q)h_{qp}-(h^{pq}{h}_{ip}-H{h}_i^q)h_{jq}\\ & ={\mathrm{\nabla }}_i{\mathrm{\nabla }}_{j}H-|h{|}^{2}h+H{h}^2;\end{array}$

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.Template:Sfnm In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.Template:Sfnm

References

Footnotes Template:Reflist Books

• Template:Wikicite
• Template:Wikicite
• Template:Wikicite

Articles

• Template:Wikicite
• Template:Wikicite
• Template:Wikicite
• Template:Wikicite