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

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
.[2] This has the consequence that

where A is the shape operator.[3] In this setting, the derivation is particularly simple:

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

  1. Simons 1968, Section 4.2.
  2. Huisken 1984, Lemma 2.1(i).
  3. Simon 1983, Lemma B.8.
  4. Huisken 1986.
  5. 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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