Bochner's formula

In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold $(M,g)$ to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.

Formal statement

If $u\colon M\rightarrow \mathbb {R}$ is a smooth function, then

\Delta {\bigg (}{\frac {|\nabla u|^{2}}{2}}{\bigg )}=\langle \nabla \Delta u,\nabla u\rangle +|\nabla ^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u)

,

where $\nabla u$ is the gradient of $u$ with respect to $g$ and ${\mbox{Ric}}$ is the Ricci curvature tensor.^[1] If $u$ is harmonic (i.e., $\Delta u=0$ , where $\Delta =\Delta _{g}$ is the Laplacian with respect to the metric $g$ ), Bochner's formula becomes

\Delta {\frac {1}{2}}|\nabla u|^{2}=|\nabla ^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u)

.

Bochner used this formula to prove the Bochner vanishing theorem.

As a corollary, if $(M,g)$ is a Riemannian manifold without boundary and $u\colon M\rightarrow \mathbb {R}$ is a smooth, compactly supported function, then

\int _{M}(\Delta u)^{2}\,d{\mbox{vol}}=\int _{M}{\Big (}|\nabla ^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u){\Big )}\,d{\mbox{vol}}

.

This immediately follows from the first identity, observing that the integral of the left-hand side vanishes (by the divergence theorem) and integrating by parts the first term on the right-hand side.

Variations and generalizations

References

^ Chow, Bennett; Lu, Peng; Ni, Lei (2006), Hamilton's Ricci flow, Graduate Studies in Mathematics, 77, Providence, RI: Science Press, New York, p. 19, ISBN 978-0-8218-4231-7, MR 2274812.

[1] Chow, Bennett; Lu, Peng; Ni, Lei (2006), Hamilton's Ricci flow, Graduate Studies in Mathematics, 77, Providence, RI: Science Press, New York, p. 19, ISBN 978-0-8218-4231-7, MR 2274812.

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