Reilly formula

In the mathematical field of Riemannian geometry, the Reilly formula is an important identity, discovered by Robert Reilly in 1977.^[1] It says that, given a smooth Riemannian manifold-with-boundary (M, g) and a smooth function u on M, one has

${\displaystyle \int _{\partial M}\left(H{\Big (}{\frac {\partial u}{\partial \nu }}{\Big )}^{2}+2{\frac {\partial u}{\partial \nu }}\Delta ^{\partial M}u+h{\big (}\nabla ^{\partial M}u,\nabla ^{\partial M}u{\big )}\right)=\int _{M}{\Big (}(\Delta u)^{2}-|\nabla \nabla u|^{2}-\operatorname {Ric} (\nabla u,\nabla u){\Big )},}$

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.^[2][3] This is often used in combination with the observation

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

with the consequence that

${\displaystyle \int _{\partial M}\left(H{\Big (}{\frac {\partial u}{\partial \nu }}{\Big )}^{2}+2{\frac {\partial u}{\partial \nu }}\Delta ^{\partial M}u+h{\big (}\nabla ^{\partial M}u,\nabla ^{\partial M}u{\big )}\right)\leq \int _{M}{\Big (}{\frac {n-1}{n}}(\Delta u)^{2}-\operatorname {Ric} (\nabla u,\nabla u){\Big )}.}$

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.^[4][5] Applications include eigenvalue estimates in spectral geometry and the study of submanifolds of constant mean curvature.