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Reilly formula

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In the mathematical field of Riemannian geometry, the Reilly formula is an important identity, discovered by Robert Reilly in 1977. It says that, given a smooth Riemannian manifold-with-boundary (M, g) and a smooth function u on M, one has

∫ ∂ M ( H ( ∂ u ∂ ν ) 2 + 2 ∂ u ∂ ν Δ ∂ M u + h ( ∇ ∂ M u , ∇ ∂ M u ) ) = ∫ M ( ( Δ u ) 2 − | ∇ ∇ u | 2 − Ric ⁡ ( ∇ u , ∇ u ) ) , {\displaystyle \int _{\partial M}\left(H{\Big (}{\frac {\partial u}{\partial \nu }}{\Big )}^{2}+2{\frac {\partial u}{\partial \nu }}\Delta ^{\partial M}u+h{\big (}\nabla ^{\partial M}u,\nabla ^{\partial M}u{\big )}\right)=\int _{M}{\Big (}(\Delta u)^{2}-|\nabla \nabla u|^{2}-\operatorname {Ric} (\nabla u,\nabla u){\Big )},}

in which h is the second fundamental form of the boundary of M, H is its mean curvature, and ν is its unit normal vector. This is often used in combination with the observation

| ∇ ∇ u | 2 = 1 n ( Δ u ) 2 + | ∇ ∇ u − 1 n ( Δ u ) g | 2 ≥ 1 n ( Δ u ) 2 , {\displaystyle |\nabla \nabla u|^{2}={\frac {1}{n}}(\Delta u)^{2}+{\Big |}\nabla \nabla u-{\frac {1}{n}}(\Delta u)g{\Big |}^{2}\geq {\frac {1}{n}}(\Delta u)^{2},}

with the consequence that

∫ ∂ M ( H ( ∂ u ∂ ν ) 2 + 2 ∂ u ∂ ν Δ ∂ M u + h ( ∇ ∂ M u , ∇ ∂ M u ) ) ≤ ∫ M ( n − 1 n ( Δ u ) 2 − Ric ⁡ ( ∇ u , ∇ u ) ) . {\displaystyle \int _{\partial M}\left(H{\Big (}{\frac {\partial u}{\partial \nu }}{\Big )}^{2}+2{\frac {\partial u}{\partial \nu }}\Delta ^{\partial M}u+h{\big (}\nabla ^{\partial M}u,\nabla ^{\partial M}u{\big )}\right)\leq \int _{M}{\Big (}{\frac {n-1}{n}}(\Delta u)^{2}-\operatorname {Ric} (\nabla u,\nabla u){\Big )}.}

This is particularly useful since one can now make use of the solvability of the Dirichlet problem for the Laplacian to make useful choices for u. Applications include eigenvalue estimates in spectral geometry and the study of submanifolds of constant mean curvature.

References

  1. Reilly 1977
  2. Chow, Lu, and Ni, section A.5
  3. Colding and Minicozzi, section 7.3
  4. Li, section 8
  5. Schoen and Yau, section III.8
  • Bennett Chow, Peng Lu, and Lei Ni. Hamilton's Ricci flow. Graduate Studies in Mathematics, 77. American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. xxxvi+608 pp. ISBN 978-0-8218-4231-7, 0-8218-4231-5
  • 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. ISBN 978-0-8218-5323-8. doi:10.1090/gsm/121
  • Peter Li. Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012. x+406 pp. ISBN 978-1-107-02064-1. doi:10.1017/CBO9781139105798 Closed access icon
  • Reilly, Robert (1977). "Applications of the Hessian operator in a Riemannian manifold". Indiana University Mathematics Journal. 26 (3): 459. doi:10.1512/iumj.1977.26.26036. ISSN 0022-2518.Free access icon
  • R. Schoen and S.-T. Yau. Lectures on differential geometry. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang, Jia Qing Zhong and Yi Chao Xu. Translated from the Chinese by Ding and S.Y. Cheng. With a preface translated from the Chinese by Kaising Tso. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. v+235 pp. ISBN 1-57146-012-8

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