An obstruction for the mean curvature of a conformal immersion ⁿ→ℝⁿ

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H H of a conformal immersion S n → R n + 1 S^n\to \mathbb {R}^{n+1} satisfies ∫ ∂ X H = 0 \int \partial _X H=0...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2007-02, Vol.135 (2), p.489-493
Main Authors: Ammann, Bernd, Humbert, Emmanuel, Ahmedou, Mohameden
Format: Article
Language:English
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Summary:We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H H of a conformal immersion S n → R n + 1 S^n\to \mathbb {R}^{n+1} satisfies ∫ ∂ X H = 0 \int \partial _X H=0 where X X is a conformal vector field on S n S^n and where the integration is carried out with respect to the Euclidean volume measure of the image. This identity is analogous to the Kazdan-Warner obstruction that appears in the problem of prescribing the scalar curvature on S n S^n inside the standard conformal class.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-06-08491-7