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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.cite-ref-footnotesimons1968section-4-2-1-0[1] 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

Δ Δ h = Hess ⁡ ⁡ H + H h 2 − − | h | 2 h , {\displaystyle \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 h2 is the symmetric 2-tensor on M given by h2

ij = gpqhiphqj.cite-ref-footnotehuisken1984lemma-2-1-i-2-0[2] This has the consequence that

1 2 Δ Δ | h | 2 = | ∇ ∇ h | 2 − − | h | 4 + ⟨ ⟨ h , Hess ⁡ ⁡ H ⟩ ⟩ + H tr ⁡ ⁡ ( A 3 ) {\displaystyle {\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.cite-ref-footnotesimon1983lemma-b-8-3-0[3] In this setting, the derivation is particularly simple:

Δ Δ h i j = ∇ ∇ p ∇ ∇ p h i j = ∇ ∇ p ∇ ∇ i h j p = ∇ ∇ i ∇ ∇ p h j p − − R p i j q h q p − − R p i p q h j q = ∇ ∇ i ∇ ∇ j H − − ( h p q h i j − − h j p h i q ) h q p − − ( h p q h i p − − H h i q ) h j q = ∇ ∇ i ∇ ∇ j H − − | h | 2 h + H h 2 ; {\displaystyle {\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.cite-ref-footnotehuisken1986-4-0[4] In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.cite-ref-footnotesimons1968section-4-2cherndo-carmokobayashi1970-5-0[5]

cite-note-footnotesimons1968section-4-2-11. ↑ Simons 1968, Section 4.2.

cite-note-footnotehuisken1984lemma-2-1-i-22. ↑ Huisken 1984, Lemma 2.1(i).

cite-note-footnotesimon1983lemma-b-8-33. ↑ Simon 1983, Lemma B.8.

cite-note-footnotehuisken1986-44. ↑ Huisken 1986.

cite-note-footnotesimons1968section-4-2cherndo-carmokobayashi1970-55. ↑ Simons 1968, Section 4.2; Chern, do Carmo & Kobayashi 1970.

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