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}}}, {{{2}}}) 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
- [math]\displaystyle{ \Delta h=\operatorname{Hess}H+Hh^2-|h|^2h, }[/math]
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 h2ij = gpqhiphqj.({{{1}}}, {{{2}}}) This has the consequence that
- [math]\displaystyle{ \frac{1}{2}\Delta|h|^2=|\nabla h|^2-|h|^4+\langle h,\operatorname{Hess}H\rangle+H\operatorname{tr}(A^3) }[/math]
where A is the shape operator.({{{1}}}, {{{2}}}) In this setting, the derivation is particularly simple:
- [math]\displaystyle{ \begin{align} \Delta h_{ij}&=\nabla^p\nabla_p h_{ij}\\ &=\nabla^p\nabla_ih_{jp}\\ &=\nabla_i\nabla^p h_{jp}-{{R^p}_{ij}}^qh_{qp}-{{R^p}_{ip}}^qh_{jq}\\ &=\nabla_i\nabla_jH-(h^{pq}h_{ij}-h_j^ph_i^q)h_{qp}-(h^{pq}h_{ip}-Hh_i^q)h_{jq}\\ &=\nabla_i\nabla_jH-|h|^2h+Hh^2; \end{align} }[/math]
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.({{{1}}}, {{{2}}}) In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.({{{1}}}, {{{2}}})
References
Footnotes
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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