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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An obstruction for the mean curvature of a conformal immersion $ S^n\to \mathbb{R}^{n+1}$


Authors: Bernd Ammann, Emmanuel Humbert and Mohameden Ould Ahmedou
Journal: Proc. Amer. Math. Soc. 135 (2007), 489-493
MSC (2000): Primary 53A27, 53A30, 35J60
Published electronically: August 28, 2006
MathSciNet review: 2255295
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Abstract: 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$ of a conformal immersion $ S^n\to \mathbb{R}^{n+1}$ satisfies $ \int \partial_X H=0$ where $ X$ is a conformal vector field on $ 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$ inside the standard conformal class.


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Additional Information

Bernd Ammann
Affiliation: Institut Élie Cartan, BP 239, Université de Nancy 1, 54506 Vandoeuvre-lès-Nancy Cedex, France
Email: ammann@iecn.u-nancy.fr

Emmanuel Humbert
Affiliation: Institut Élie Cartan, BP 239, Université de Nancy 1, 54506 Vandoeuvre-lès-Nancy Cedex, France
Email: humbert@iecn.u-nancy.fr

Mohameden Ould Ahmedou
Affiliation: Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: ahmedou@analysis.mathematik.uni-tuebingen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08491-7
PII: S 0002-9939(06)08491-7
Received by editor(s): June 28, 2005
Received by editor(s) in revised form: September 6, 2005
Published electronically: August 28, 2006
Communicated by: Mikhail Shubin
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.