Reilly formula

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

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

with the consequence that
 * $$\int_{\partial M}\left(H\Big(\frac{\partial u}{\partial\nu}\Big)^2+2\frac{\partial u}{\partial\nu}\Delta^{\partial M}u+h\big(\nabla^{\partial M}u,\nabla^{\partial M}u\big)\right)\leq\int_M \Big(\frac{n-1}{n}(\Delta u)^2-\operatorname{Ric}(\nabla u,\nabla u)\Big).$$

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.