Simons' formula

Last updated April 15, 2022

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.^[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

\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 $h 2 ij = g pq h ip h qj$ .^[2] 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.^[3] 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.^[4] In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.^[5]

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References

Footnotes

↑ Simons 1968, Section 4.2.
↑ Huisken 1984, Lemma 2.1(i).
↑ Simon 1983, Lemma B.8.
↑ Huisken 1986.
↑ Simons 1968, Section 4.2; Chern, do Carmo & Kobayashi 1970.

Books

Tobias Holck Colding and William P. Minicozzi, II. A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. ISBN 978-0-8218-5323-8
Enrico Giusti. Minimal surfaces and functions of bounded variation. Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. xii+240 pp. ISBN 0-8176-3153-4
Leon Simon. Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983. vii+272 pp. ISBN 0-86784-429-9

Articles

S.S. Chern, M. do Carmo, and S. Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length. Functional Analysis and Related Fields (1970), 59–75. Proceedings of a Conference in honor of Professor Marshall Stone, held at the University of Chicago, May 1968. Springer, New York. Edited by Felix E. Browder. doi:10.1007/978-3-642-48272-4_2
Gerhard Huisken. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 20 (1984), no. 1, 237–266. doi:10.4310/jdg/1214438998
Gerhard Huisken. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84 (1986), no. 3, 463–480. doi:10.1007/BF01388742
James Simons. Minimal varieties in Riemannian manifolds. Ann. of Math. (2) 88 (1968), 62–105. doi:10.2307/1970556

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[FOOTNOTESimons1968Section_4.2-1] Simons 1968, Section 4.2.

[FOOTNOTEHuisken1984Lemma_2.1(i)-2] Huisken 1984, Lemma 2.1(i).

[FOOTNOTESimon1983Lemma_B.8-3] Simon 1983, Lemma B.8.

[FOOTNOTEHuisken1986-4] Huisken 1986.

[FOOTNOTESimons1968Section_4.2Cherndo_CarmoKobayashi1970-5] Simons 1968, Section 4.2; Chern, do Carmo & Kobayashi 1970.

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