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]
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. ISBN978-0-8218-5323-8
Enrico Giusti. Minimal surfaces and functions of bounded variation. Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. xii+240 pp. ISBN0-8176-3153-4
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. ISBN0-86784-429-9
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
March 15, 2024
simons, formula, mathematical, field, differential, geometry, simons, formula, also, known, simons, identity, some, variants, simons, inequality, fundamental, equation, study, minimal, submanifolds, discovered, james, simons, 1968, viewed, formula, laplacian, . 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 D h Hess H H h 2 h 2 h displaystyle 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 h2 is the symmetric 2 tensor on M given by h2ij gpqhiphqj 2 This has the consequence that 1 2 D h 2 h 2 h 4 h Hess H H tr A 3 displaystyle 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 D h i j p p h i j p i h j p i p h j p R p i j q h q p R p i p q h j q i j H h p q h i j h j p h i q h q p h p q h i p H h i q h j q i j H h 2 h H h 2 displaystyle begin aligned Delta h ij amp nabla p nabla p h ij amp nabla p nabla i h jp amp nabla i nabla p h jp R p ij q h qp R p ip q h jq amp 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 amp 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 editFootnotes 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 amp 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 Birkhauser 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 9Articles 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 nbsp 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 nbsp Gerhard Huisken Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature Invent Math 84 1986 no 3 463 480 doi 10 1007 BF01388742 nbsp James Simons Minimal varieties in Riemannian manifolds Ann of Math 2 88 1968 62 105 doi 10 2307 1970556 nbsp Retrieved from https en wikipedia org w index php title Simons 27 formula amp oldid 1082718431, wikipedia, wiki, book, books, library,