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

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.