Simons' formula
{{short description|Mathematical 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.{{sfnm|1a1=Simons|1y=1968|1loc=Section 4.2}} 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 {{mvar|M}} of Euclidean space, the formula asserts that
:
where, relative to a local choice of unit normal vector field, {{mvar|h}} is the second fundamental form, {{mvar|H}} is the mean curvature, and {{math|h2}} is the symmetric 2-tensor on {{mvar|M}} given by {{math|h{{su|b=ij|p=2}} {{=}} gpqhiphqj}}.{{sfnm|1a1=Huisken|1y=1984|1loc=Lemma 2.1(i)}}
This has the consequence that
:
where {{mvar|A}} is the shape operator.{{sfnm|1a1=Simon|1y=1983|1loc=Lemma B.8}} In this setting, the derivation is particularly simple:
:
\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}
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.{{sfnm|1a1=Huisken|1y=1986}} In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.{{sfnm|1a1=Simons|1y=1968|1loc=Section 4.2|2a1=Chern|2a2=do Carmo|2a3=Kobayashi|2y=1970}}
References
Footnotes
{{reflist|30em}}
Books
- {{wikicite|ref={{sfnRef|Colding|Minicozzi|2011}}|reference=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}}}}
- {{wikicite|ref={{sfnRef|Giusti|1984}}|reference=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}}}}
- {{wikicite|ref={{sfnRef|Simon|1983}}|reference=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
- {{wikicite|ref={{sfnRef|Chern|do Carmo|Kobayashi|1970}}|reference=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}} {{closed access}}}}
- {{wikicite|ref={{sfnRef|Huisken|1984}}|reference=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}} {{free access}}}}
- {{wikicite|ref={{sfnRef|Huisken|1986}}|reference=Gerhard Huisken. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84 (1986), no. 3, 463–480. {{doi|10.1007/BF01388742}} {{closed access}}}}
- {{wikicite|ref={{sfnRef|Simons|1968}}|reference=James Simons. Minimal varieties in Riemannian manifolds. Ann. of Math. (2) 88 (1968), 62–105. {{doi|10.2307/1970556}} {{closed access}}}}