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Bochner's formula

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

Formal statement

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

1 2 Δ | ∇ u | 2 = g ( ∇ Δ u , ∇ u ) + | ∇ 2 u | 2 + Ric ( ∇ u , ∇ u ) {\displaystyle {\tfrac {1}{2}}\Delta |\nabla u|^{2}=g(\nabla \Delta u,\nabla u)+|\nabla ^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u)} ,

where ∇ u {\displaystyle \nabla u} is the gradient of u {\displaystyle u} with respect to g {\displaystyle g} , ∇ 2 u {\displaystyle \nabla ^{2}u} is the Hessian of u {\displaystyle u} with respect to g {\displaystyle g} and Ric {\displaystyle {\mbox{Ric}}} is the Ricci curvature tensor. If u {\displaystyle u} is harmonic (i.e., Δ u = 0 {\displaystyle \Delta u=0} , where Δ = Δ g {\displaystyle \Delta =\Delta _{g}} is the Laplacian with respect to the metric g {\displaystyle g} ), Bochner's formula becomes

1 2 Δ | ∇ u | 2 = | ∇ 2 u | 2 + Ric ( ∇ u , ∇ u ) {\displaystyle {\tfrac {1}{2}}\Delta |\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 ) {\displaystyle (M,g)} is a Riemannian manifold without boundary and u : M → R {\displaystyle u\colon M\rightarrow \mathbb {R} } is a smooth, compactly supported function, then

∫ M ( Δ u ) 2 d vol = ∫ M ( | ∇ 2 u | 2 + Ric ( ∇ u , ∇ u ) ) d vol {\displaystyle \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

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