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I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only).

For example I will need the following formula: $$\int\limits_\Omega |\Delta f|^2 -|\mathrm{Hess}f|^2 +\langle\mathrm{Ric}(\nabla f),\nabla f\rangle =\int\limits_{\partial\Omega} H\cdot|\nabla f|^2,$$ where $H$ denotes mean curvature of $\partial \Omega$ and $f$ vanish on the boundary, but I will also need its analog for 1-forms and yet general boundary condition.

So far I found "Spin geometry" by Lawson and Michelsohn --- based on II/§5, it is easy to derive any formula I need (but I am sure someone did it somewhere).

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    $\begingroup$ What is a relative Bochner formula? IIRC, Wu's "The Bochner Technique in Differential Geometry" (bookstore.ams.org/ctm-6) states the Bochner-Weitzenböck formulas for 1-forms and functions as corollaries. $\endgroup$
    – M.G.
    Jun 27, 2018 at 21:49
  • $\begingroup$ @M.G. I mean for domains with boundary. $\endgroup$
    – Anton Petrunin
    Jun 27, 2018 at 22:15
  • $\begingroup$ @M.G. No, I did not find it in this book. $\endgroup$
    – Anton Petrunin
    Jun 28, 2018 at 9:54
  • $\begingroup$ Anton, it appears you are having something different in mind than what I thought initially, apologies. I was thinking of the following (stated on p.307 in Wu's book more or less): if $\psi$ is a harmonic 1-form, then $-\Delta||\psi||^2 = 2\sum_{k=1}^n||\nabla_{X_k}\psi||^2 + 2\operatorname{Ric}(\psi^\#,\psi^\#)$, where $\{X_k:1\leq k\leq n\}$ is a frame field (sorry, notations are from my own notes, so slightly different than in Wu's book). Weird, I was sure there was an explicit statement for harmonic 0-forms, but now I see there is not (but it follows from WFII). $\endgroup$
    – M.G.
    Jun 28, 2018 at 13:14
  • $\begingroup$ Did you check Riemannian Geometry by Petersen? If I correctly remember he discusses the Bochner and the Bochner-Weitzenböck formulas quite extensively. $\endgroup$
    – Piotr Hajlasz
    Jul 2, 2018 at 23:35

1 Answer 1

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Here's a reference to a paper; Theorem 3 gives a general formula for differential forms with no constraints on the boundary values. I'm not sure if this shows up in a book yet.

Raulot, S.; Savo, A. A Reilly formula and eigenvalue estimates for differential forms. J. Geom. Anal. 21 (2011), no. 3, 620--640. MR2810846.

Here is a direct link to the article.

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  • $\begingroup$ I fixed the MR link. Are you looking for a slightly different version of the Reilly formula than that presented in this reference? $\endgroup$
    – Jeffrey Case
    Jul 1, 2018 at 17:24
  • $\begingroup$ It seems to be all I need, except the notations are different from those I used to. Also, I feel more comfortable to refer to a book; say by now I would better use "Spin Geometry" of Lawson and Michelsohn. $\endgroup$
    – Anton Petrunin
    Jul 1, 2018 at 21:05

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