In the mathematical field of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...

, the Reilly formula is an important identity, discovered by Robert Reilly in 1977. It says that, given a smooth Riemannian manifold-with-boundary and a

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

on , one has :

\int_\left(H\Big(\frac\Big)^2+2\frac\Delta^u+h\big(\nabla^u,\nabla^u\big)\right)=\int_M \Big((\Delta u)^2-, \nabla\nabla u, ^2-\operatorname(\nabla u,\nabla u)\Big),

in which is the

second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...

of the boundary of , is its

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. T ...

, and is its unit normal vector. This is often used in combination with the observation :

, \nabla\nabla u, ^2=\frac(\Delta u)^2+\Big, \nabla\nabla u-\frac(\Delta u)g\Big, ^2\geq\frac(\Delta u)^2,

with the consequence that :

\int_\left(H\Big(\frac\Big)^2+2\frac\Delta^u+h\big(\nabla^u,\nabla^u\big)\right)\leq\int_M \Big(\frac(\Delta u)^2-\operatorname(\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 .Schoen and Yau, section III.8 Applications include eigenvalue estimates in

spectral geometry Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifol ...

and the study of submanifolds of constant mean curvature.

References

* 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. * 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. . * Peter Li. ''Geometric analysis.'' Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012. x+406 pp. . * * 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

External links

In Memoriam Robert Cunningham Reilly
Differential geometry