A Pohozaev type identity for Dirac operators

Bernd Ammann (1) , Emmanuel Humbert (1) , Mohameden Ould Ahmedou (2)

1 IECN - Institut Élie Cartan de Nancy
2 TüBINGEN - Mathematik Universität Tübingen

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 $S^n\to \mR^{n+1}$ satisfies $\int \pa_X H=0$ 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.

Mots clés: Dirac operator; pohozaev identity

Domaines: Géométrie différentielle [math.DG]

Bernd Ammann, Emmanuel Humbert, Mohameden Ould Ahmedou. A Pohozaev type identity for Dirac operators. Proceedings of the American Mathematical Society, 2007, 135, pp.489-493. ⟨hal-00091168⟩