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.[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

{\frac {1}{2}}\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]

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.