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An obstruction for the mean curvature of a conformal immersion S^{n}-> R^{n+1}

Bernd Ammann·Emmanuel Humbert·Mohameden Ould Ahmedou
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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}-> 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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Bernd Ammann, Emmanuel Humbert, Mohameden Ould Ahmedou. 2005-06-28. An obstruction for the mean curvature of a conformal immersion S^{n}-> R^{n+1}. https://arxiv.org/abs/math/0506568

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